![]() Thus wewould reject the hypothesis that the true slope is zero (or negative)and conclude that an increase in income raises foodexpenditures. If the t-statistic reported for the slopecoefficient is enough larger than zero, we can conclude that it isvery improbable that the sample we have would have come from adata-generating process in which the true slope is zero. (For more details about the t-distribution, look at Special Topic: The t-distribution. Under conditions that are often assumed to be true, the probability distribution of this ratio will be a t-distribution. This t-statistic is reported by most regression software programs. The resulting ratio tells us how many standard-error units the coefficient is away from zero. To perform this test,we calculate a t-statistic by dividing the estimated coefficient by its standard error. The most common hypothesis test in econometrics is the t-test of the null hypothesis that a coefficient equals zero. Choosing a significance levelof 0.05 for our test of slope means that we reject the hypothesis ofa zero (or negative) slope if sample evidence of positive slope asstrong as our would happen due to random variation less than 5% ofthe time if the slope were truly zero. Conventional choices for the level ofsignificance are 0.10, 0.05, and 0.01. What threshold should we set for how small thep-value must be in order to satisfy us that the null hypothesis isfalse? This threshold is called the level of significance ofthe statistical test. We then pose the following question: "If the trueslope is zero, how unlikely is it that we would find a sample of datathat presents evidence for a positive slope that it as convincing asthe sample we have actually observed?" If that probability (oftencalled the probability value or p-value) is smallenough, we reject the null hypothesis that the slope is zero ornegative and conclude that the true slope is positive. We can use the coefficient estimates and theirstandard errors to try to answer the question: "How confident can webe (based on our sample estimate that the slope is 0.23) that theslope is not negative?" The way we go about answering this questionis to assume as a null hypothesis that the actual slope is zero. If, as we sometimes assume, the coefficientestimators can be assumed to follow a normal distribution, then theestimated coefficient ( b 1 or b 2)will lie within one standard error distance of the true parametervalue ( B 1 or B 2) about 68% of thetime and within a range of two standard error distances above andbelow the true value about 95% of the time. They are estimates of the standard deviation of the estimator which, as was explained in the discussion of the Previous Topic: Sampling Variation, measures how widely the estimates are distributed around the mean. These standard errors are reported by regression software packages. Linear regression measures the degree of confidence we may have about our estimates through the standard errors of our estimated coefficients. If the pointsare widely scattered around the line, then it is more plausible thata line with zero or negative slope could fit the data almost as wellas our estimated line with slope of 0.23. If all of the data points in the samplelie very close to a line with slope 0.23, then we may feel quiteconfident that 0.23 is a good estimate of the slope. An estimate of 0.05 would give us less confidence.The second factor is how clearly the data in the sample are tellingus that the slope is 0.23. ![]() Other things beingequal, an estimate of 0.5 would make us more confident that B 2 is positive than the estimate we of 0.23 that weactually obtained. Let's focus on this final question: How sure canwe be that the actual slope ( B 2) is not zero ornegative? How convincing is the evidence for a positive slope fromour sample? Two factors enter into our assessment of this question.The first is how "positive" our slope estimate is. How confident can we be, based on oureconometric regression, that the true value of the slope is notlarger than 0.30? Or smaller than 0.15? Or zero or negative? But we know that this is just an estimateand that the true value of the slope ( B 2) isprobably not exactly 0.23. That means that 0.23 is ourbest single guess at the amount of an additional dollar of incomethat will be spent on food. ![]() Suppose that we have run a linear regression offood expenditures on income and estimated the slope of the regressionline ( b 2) to be 0.23. Testing Hypotheses about Regression Coefficients REDSPOTS Testing Hypotheses about RegressionCoefficients
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