{"pageProps":{"post":{"coverImage":{"_type":"image","asset":{"_type":"reference","_ref":"image-198f8c63b3d453e3181f438fa63bb6ea0193e987-600x654-webp"}},"slug":"granger-causality","content":{"compiledSource":"/*@jsxRuntime automatic @jsxImportSource react*/\nconst {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0];\nconst {useMDXComponents: _provideComponents} = arguments[0];\nfunction _createMdxContent(props) {\n const _components = Object.assign({\n p: \"p\",\n h2: \"h2\",\n a: \"a\",\n span: \"span\",\n em: \"em\",\n img: \"img\",\n strong: \"strong\",\n math: \"math\",\n semantics: \"semantics\",\n mrow: \"mrow\",\n mi: \"mi\",\n annotation: \"annotation\",\n ol: \"ol\",\n li: \"li\",\n mo: \"mo\",\n msub: \"msub\",\n mn: \"mn\",\n munder: \"munder\",\n mtable: \"mtable\",\n mtr: \"mtr\",\n mtd: \"mtd\",\n mstyle: \"mstyle\",\n div: \"div\",\n msup: \"msup\",\n mover: \"mover\",\n msubsup: \"msubsup\",\n mfrac: \"mfrac\",\n table: \"table\",\n thead: \"thead\",\n tr: \"tr\",\n th: \"th\",\n tbody: \"tbody\",\n td: \"td\",\n pre: \"pre\",\n code: \"code\",\n munderover: \"munderover\"\n }, _provideComponents(), props.components);\n return _jsxs(_Fragment, {\n children: [_jsx(_components.p, {\n children: \"This article discusses causal illusions as a form of cognitive bias and explores the use of Granger causality to detect causal structures in time series. It is common practice to analyse (linear) structure, estimate linear models and perform forecasts based on single stationary time series. However, the world does not consist of independent stochastic processes. In accordance with general equilibrium theory, economists usually assume that everything depends on everything else. Therefore, it is important to understand and quantify the (causal) relationships between different time series.\"\n }), \"\\n\", _jsxs(_components.h2, {\n id: \"epiphenomena\",\n children: [_jsx(_components.a, {\n className: \"anchor\",\n href: \"#epiphenomena\",\n children: _jsx(_components.span, {\n className: \"icon icon-link\"\n })\n }), \"Epiphenomena\"]\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"Epiphenomena is a class of causal illusions where the direction of causal relationships is ambiguous. For example, when you spend time on the bridge of a ship with a large compass in front, you can easily develop the impression that the compass is directing the ship rather than merely reflecting its direction. Here is an image that perfectly illustrates the point that \", _jsx(_components.em, {\n children: \"correlation\"\n }), \" is not \", _jsx(_components.em, {\n children: \"causation\"\n }), \":\"]\n }), \"\\n\", _jsx(_components.p, {\n children: _jsx(_components.img, {\n src: \"https://cdn.sanity.io/images/xj3nsfqa/production/198f8c63b3d453e3181f438fa63bb6ea0193e987-600x654.webp\",\n alt: \"image\"\n })\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"Nassim Nicholas Taleb explored this concept in his book \", _jsx(_components.em, {\n children: \"Antifragile\"\n }), \" to highlight the causal illusion that universities generate wealth in society. He presented a miscellany of evidence which suggests that classroom education does not lead to wealth as much as it comes from wealth (an \", _jsx(_components.strong, {\n children: \"epiphenomenon\"\n }), \"). Taleb proposes that antifragile risk-taking is largely responsible for innovation and growth instead of education and formal, organized research. However, it does not mean that theories and research play no role, but rather shows that we are fooled by randomness into overestimating the role of good-sounding ideas. Because of cognitive biases, historians are prone to epiphenomena and other illusions of cause and effect.\"]\n }), \"\\n\", _jsx(_components.p, {\n children: \"We can debunk epiphenomena in the cultural discourse and consciousness by looking at the sequence of events and checking their order or occurrences. This method is refined by Clive Granger who developed a rigorously scientific approach that can be used to establish causation by looking at time series sequences and measuring the \\\"Granger cause\\\".\"\n }), \"\\n\", _jsxs(_components.h2, {\n id: \"granger-causality\",\n children: [_jsx(_components.a, {\n className: \"anchor\",\n href: \"#granger-causality\",\n children: _jsx(_components.span, {\n className: \"icon icon-link\"\n })\n }), \"Granger causality\"]\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"In the following, we present the definition of Granger causality and the different possibilities of causal events resulting from it. Consider two weakly stationary time series \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \":\"]\n }), \"\\n\", _jsxs(_components.ol, {\n children: [\"\\n\", _jsxs(_components.li, {\n children: [\"\\n\", _jsxs(_components.p, {\n children: [_jsx(_components.em, {\n children: \"Granger Causality\"\n }), \": \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" is (simply) Granger causal to \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" if and only if the application of an optimal linear prediction function leads to a smaller forecast error variance on the future value of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" if current and past values of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" are used.\"]\n }), \"\\n\"]\n }), \"\\n\", _jsxs(_components.li, {\n children: [\"\\n\", _jsxs(_components.p, {\n children: [_jsx(_components.em, {\n children: \"Instantaneous Granger Causality\"\n }), \": \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" is instantaneously Granger causal to \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" if and only if the application of an optimal linear prediction function leads to a smaller forecast error variance on the future value of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" if the future value of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" is used in addition to the current and past values of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \".\"]\n }), \"\\n\"]\n }), \"\\n\", _jsxs(_components.li, {\n children: [\"\\n\", _jsxs(_components.p, {\n children: [_jsx(_components.em, {\n children: \"Feedback\"\n }), \": There is feedback between \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" if \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" is causal to \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" is causal to \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \". Feedback is only defined for the case of simple causal relations.\"]\n }), \"\\n\"]\n }), \"\\n\"]\n }), \"\\n\", _jsxs(_components.h2, {\n id: \"hypothesis-testing-in-general-linear-models\",\n children: [_jsx(_components.a, {\n className: \"anchor\",\n href: \"#hypothesis-testing-in-general-linear-models\",\n children: _jsx(_components.span, {\n className: \"icon icon-link\"\n })\n }), \"Hypothesis testing in general linear models\"]\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"Consider the general linear model \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n mathvariant: \"bold\",\n children: \"Y\"\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mi, {\n children: \"X\"\n }), _jsx(_components.mi, {\n mathvariant: \"bold-italic\",\n children: \"β\"\n }), _jsx(_components.mo, {\n children: \"+\"\n }), _jsx(_components.mi, {\n children: \"ε\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\mathbf{Y} = X \\\\boldsymbol{\\\\beta} + \\\\varepsilon\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6861em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathbf\",\n style: {\n marginRight: \"0.02875em\"\n },\n children: \"Y\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord boldsymbol\",\n style: {\n marginRight: \"0.03403em\"\n },\n children: \"β\"\n })\n })\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"+\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"ε\"\n })]\n })]\n })]\n })\n }), \". In hypothesis testing, we want to know whether certain variables influence the result. If, say, the variable \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"x\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x_1\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.5806em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" does not influence \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n mathvariant: \"bold\",\n children: \"Y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\mathbf{Y}\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6861em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathbf\",\n style: {\n marginRight: \"0.02875em\"\n },\n children: \"Y\"\n })]\n })\n })]\n })\n }), \", then we must have \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\beta_1 = 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \". So the goal is to test the hypothesis \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"H\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n children: \":\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"H_0: \\\\beta_1 = 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.08125em\"\n },\n children: \"H\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0813em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \":\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \" versus \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"H\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \":\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n mathvariant: \"normal\",\n children: \"≠\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"H_1: \\\\beta_1 \\\\neq 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.08125em\"\n },\n children: \"H\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0813em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \":\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsxs(_components.span, {\n className: \"mrel\",\n children: [_jsx(_components.span, {\n className: \"mrel\",\n children: _jsx(_components.span, {\n className: \"mord vbox\",\n children: _jsx(_components.span, {\n className: \"thinbox\",\n children: _jsxs(_components.span, {\n className: \"rlap\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"inner\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mrel\",\n children: \"\"\n })\n })\n }), _jsx(_components.span, {\n className: \"fix\"\n })]\n })\n })\n })\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \". We will tackle a more general case, where \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"β\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\beta\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n })]\n })\n })]\n })\n }), \" can be split into two vectors \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\beta_0\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\beta_1\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \", and we test if \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\beta_1\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05278em\"\n },\n children: \"β\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0528em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" is zero.\"]\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"Suppose \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: _jsxs(_components.munder, {\n children: [_jsx(_components.mo, {\n children: _jsx(_components.mi, {\n children: \"X\"\n })\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"n\"\n }), _jsx(_components.mo, {\n children: \"×\"\n }), _jsx(_components.mi, {\n children: \"p\"\n })]\n })]\n })\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsx(_components.mi, {\n children: _jsxs(_components.munder, {\n children: [_jsx(_components.mo, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"X\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"n\"\n }), _jsx(_components.mo, {\n children: \"×\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"p\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })]\n })]\n })\n }), _jsx(_components.mi, {\n children: _jsxs(_components.munder, {\n children: [_jsx(_components.mo, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"X\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n })\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"n\"\n }), _jsx(_components.mo, {\n children: \"×\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsx(_components.mi, {\n children: \"p\"\n }), _jsx(_components.mo, {\n children: \"−\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"p\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n })]\n })]\n })\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\underset{n \\\\times p}{X} = \\\\left(\\\\underset{n \\\\times p_0}{X_0} \\\\underset{n \\\\times\\\\left(p - p_0\\\\right)}{X_1}\\\\right)\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.5194em\",\n verticalAlign: \"-0.8361em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mop op-limits\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.6833em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.4em\",\n marginLeft: \"0em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"n\"\n }), _jsx(_components.span, {\n className: \"mbin mtight\",\n children: \"×\"\n }), _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"p\"\n })]\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n children: _jsx(_components.span, {\n className: \"mop mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8361em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"3em\",\n verticalAlign: \"-1.25em\"\n }\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size4\",\n children: \"(\"\n })\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mop op-limits\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.6833em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.25em\",\n marginLeft: \"0em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"n\"\n }), _jsx(_components.span, {\n className: \"mbin mtight\",\n children: \"×\"\n }), _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"p\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3173em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.357em\",\n marginLeft: \"0em\",\n marginRight: \"0.0714em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.5em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size3 size1 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.143em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n children: _jsx(_components.span, {\n className: \"mop\",\n children: _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0785em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.9861em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mop op-limits\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.6833em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.159em\",\n marginLeft: \"0em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"n\"\n }), _jsx(_components.span, {\n className: \"mbin mtight\",\n children: \"×\"\n }), _jsxs(_components.span, {\n className: \"minner mtight\",\n children: [_jsx(_components.span, {\n className: \"mopen mtight delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"mtight\",\n children: \"(\"\n })\n }), _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"p\"\n }), _jsx(_components.span, {\n className: \"mbin mtight\",\n children: \"−\"\n }), _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"p\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3173em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.357em\",\n marginLeft: \"0em\",\n marginRight: \"0.0714em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.5em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size3 size1 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.143em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose mtight delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"mtight\",\n children: \")\"\n })\n })]\n })]\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n children: _jsx(_components.span, {\n className: \"mop\",\n children: _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0785em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"1.116em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size4\",\n children: \")\"\n })\n })]\n })]\n })]\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n mathvariant: \"bold-italic\",\n children: \"β\"\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsxs(_components.mtable, {\n rowspacing: \"0.16em\",\n columnalign: \"center\",\n columnspacing: \"1em\",\n children: [_jsx(_components.mtr, {\n children: _jsx(_components.mtd, {\n children: _jsx(_components.mstyle, {\n scriptlevel: \"0\",\n displaystyle: \"false\",\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n mathvariant: \"bold-italic\",\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })\n })\n })\n }), _jsx(_components.mtr, {\n children: _jsx(_components.mtd, {\n children: _jsx(_components.mstyle, {\n scriptlevel: \"0\",\n displaystyle: \"false\",\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n mathvariant: \"bold-italic\",\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n })\n })\n })\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\boldsymbol{\\\\beta} = \\\\left(\\\\begin{array}{c}\\\\boldsymbol{\\\\beta}_0 \\\\\\\\ \\\\boldsymbol{\\\\beta}_1\\\\end{array}\\\\right)\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8889em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord boldsymbol\",\n style: {\n marginRight: \"0.03403em\"\n },\n children: \"β\"\n })\n })\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"2.4em\",\n verticalAlign: \"-0.95em\"\n }\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \"(\"\n })\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsxs(_components.span, {\n className: \"mtable\",\n children: [_jsx(_components.span, {\n className: \"arraycolsep\",\n style: {\n width: \"0.5em\"\n }\n }), _jsx(_components.span, {\n className: \"col-align-c\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"1.45em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-3.61em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord boldsymbol\",\n style: {\n marginRight: \"0.03403em\"\n },\n children: \"β\"\n })\n })\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.207em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.4559em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2441em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-2.41em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord boldsymbol\",\n style: {\n marginRight: \"0.03403em\"\n },\n children: \"β\"\n })\n })\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.207em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.4559em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2441em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.95em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"arraycolsep\",\n style: {\n width: \"0.5em\"\n }\n })]\n })\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \")\"\n })\n })]\n })]\n })]\n })]\n })\n }), \", where \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n mathvariant: \"normal\",\n children: \"rank\"\n }), _jsx(_components.mo, {\n children: \"⁡\"\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \"(\"\n }), _jsx(_components.mi, {\n children: \"X\"\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \")\"\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mi, {\n children: \"p\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\operatorname{rank}(X)= p\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsx(_components.span, {\n className: \"mop\",\n children: _jsx(_components.span, {\n className: \"mord mathrm\",\n children: \"rank\"\n })\n }), _jsx(_components.span, {\n className: \"mopen\",\n children: \"(\"\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n }), _jsx(_components.span, {\n className: \"mclose\",\n children: \")\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"p\"\n })]\n })]\n })]\n })\n }), \", \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n mathvariant: \"normal\",\n children: \"rank\"\n }), _jsx(_components.mo, {\n children: \"⁡\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"X\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"p\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\operatorname{rank}\\\\left(X_0\\\\right) = p_0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsx(_components.span, {\n className: \"mop\",\n children: _jsx(_components.span, {\n className: \"mord mathrm\",\n children: \"rank\"\n })\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: \"(\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07847em\"\n },\n children: \"X\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0785em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: \")\"\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"p\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })]\n })]\n })\n }), \". We want to test \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"H\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n children: \":\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n mathvariant: \"bold-italic\",\n children: \"β\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"H_0: \\\\boldsymbol{\\\\beta}_1 = 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.08125em\"\n },\n children: \"H\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0813em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \":\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.9386em\",\n verticalAlign: \"-0.2441em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n 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\"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"1\"\n })]\n })]\n })]\n })\n }), \". An F test is applied to test the null hypothesis, \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"H\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n children: \":\"\n }), _jsxs(_components.msubsup, {\n children: [_jsx(_components.mi, {\n children: \"α\"\n }), _jsx(_components.mn, {\n children: \"12\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.msubsup, {\n children: [_jsx(_components.mi, {\n children: \"α\"\n }), _jsx(_components.mn, {\n children: \"12\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mo, {\n children: \"⋯\"\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.msubsup, {\n children: [_jsx(_components.mi, {\n children: \"α\"\n }), _jsx(_components.mn, {\n children: \"12\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"k\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"H_0: \\\\alpha^{1}_{12} = \\\\alpha^{1}_{12} = \\\\cdots = \\\\alpha^{k_2}_{12} = 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.08125em\"\n },\n children: \"H\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0813em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \":\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.0622em\",\n verticalAlign: \"-0.2481em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.0037em\"\n },\n children: \"α\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8141em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.4519em\",\n marginLeft: \"-0.0037em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"12\"\n })\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.063em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2481em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.0622em\",\n verticalAlign: \"-0.2481em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.0037em\"\n },\n children: \"α\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8141em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.4519em\",\n marginLeft: \"-0.0037em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"12\"\n })\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.063em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2481em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.3669em\"\n }\n }), _jsx(_components.span, {\n className: \"minner\",\n children: \"⋯\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.1973em\",\n verticalAlign: \"-0.2663em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.0037em\"\n },\n children: \"α\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.931em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.4337em\",\n marginLeft: \"-0.0037em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"12\"\n })\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.1449em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n style: {\n marginRight: \"0.03148em\"\n },\n children: \"k\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3173em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.357em\",\n marginLeft: \"-0.0315em\",\n marginRight: \"0.0714em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.5em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size3 size1 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.143em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2663em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \". By changing \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" and \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \", it can be tested whether a simple causal relation from \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"y\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"y\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.625em\",\n verticalAlign: \"-0.1944em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"y\"\n })]\n })\n })]\n })\n }), \" to \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"x\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"x\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"x\"\n })]\n })\n })]\n })\n }), \" exists. There is a feedback relation if the null hypothesis is rejected in both directions. To test whether there is instantaneous causality, we finally set \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"k\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"k_0 = 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8444em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03148em\"\n },\n children: \"k\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0315em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \" and perform a t or F test for the null hypothesis \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: 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There is a trade-off: the more lagged values we include, the better the influence of this variable can be captured. This argues for a high maximal lag. 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