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Chapter-4.pdf
1.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 4 Multiple Regression Analysis: Inference Wooldridge: Introductory Econometrics: A Modern Approach, 5e
2.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Statistical inference in the regression model Hypothesis tests about population parameters Construction of confidence intervals Sampling distributions of the OLS estimators The OLS estimators are random variables We already know their expected values and their variances However, for hypothesis tests we need to know their distribution In order to derive their distribution we need additional assumptions Assumption about distribution of errors: normal distribution Multiple Regression Analysis: Inference
3.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Assumption MLR.6 (Normality of error terms) independently of It is assumed that the unobserved factors are normally distributed around the population regression function. The form and the variance of the distribution does not depend on any of the explanatory variables. It follows that: Multiple Regression Analysis: Inference
4.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Discussion of the normality assumption The error term is the sum of âmanyâ different unobserved factors Sums of independent factors are normally distributed (CLT) Problems: âą How many different factors? Number large enough? âą Possibly very heterogenuous distributions of individual factors âą How independent are the different factors? The normality of the error term is an empirical question At least the error distribution should be âcloseâ to normal In many cases, normality is questionable or impossible by definition Multiple Regression Analysis: Inference
5.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Discussion of the normality assumption (cont.) Examples where normality cannot hold: âą Wages (nonnegative; also: minimum wage) âą Number of arrests (takes on a small number of integer values) âą Unemployment (indicator variable, takes on only 1 or 0) In some cases, normality can be achieved through transformations of the dependent variable (e.g. use log(wage) instead of wage) Under normality, OLS is the best (even nonlinear) unbiased estimator Important: For the purposes of statistical inference, the assumption of normality can be replaced by a large sample size Multiple Regression Analysis: Inference
6.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Terminology Theorem 4.1 (Normal sampling distributions) Under assumptions MLR.1 â MLR.6: The estimators are normally distributed around the true parameters with the variance that was derived earlier The standardized estimators follow a standard normal distribution âGauss-Markov assumptionsâ âClassical linear model (CLM) assumptionsâ Multiple Regression Analysis: Inference
7.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing hypotheses about a single population parameter Theorem 4.1 (t-distribution for standardized estimators) Null hypothesis (for more general hypotheses, see below) Under assumptions MLR.1 â MLR.6: If the standardization is done using the estimated standard deviation (= standard error), the normal distribution is replaced by a t-distribution The population parameter is equal to zero, i.e. after controlling for the other independent variables, there is no effect of xj on y Note: The t-distribution is close to the standard normal distribution if n-k-1 is large. Multiple Regression Analysis: Inference
8.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. t-statistic (or t-ratio) Distribution of the t-statistic if the null hypothesis is true Goal: Define a rejection rule so that, if it is true, H0 is rejected only with a small probability (= significance level, e.g. 5%) The t-statistic will be used to test the above null hypothesis. The farther the estimated coefficient is away from zero, the less likely it is that the null hypothesis holds true. But what does âfarâ away from zero mean? This depends on the variability of the estimated coefficient, i.e. its standard deviation. The t-statistic measures how many estimated standard deviations the estimated coefficient is away from zero. Multiple Regression Analysis: Inference
9.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Test against . Reject the null hypothesis in favour of the alternative hypothesis if the estimated coef- ficient is âtoo largeâ (i.e. larger than a criti- cal value). Construct the critical value so that, if the null hypothesis is true, it is rejected in, for example, 5% of the cases. In the given example, this is the point of the t- distribution with 28 degrees of freedom that is exceeded in 5% of the cases. ! Reject if t-statistic greater than 1.701 Multiple Regression Analysis: Inference Testing against one-sided alternatives (greater than zero)
10.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Wage equation Test whether, after controlling for education and tenure, higher work experience leads to higher hourly wages Standard errors Test against . One would either expect a positive effect of experience on hourly wage or no effect at all. Multiple Regression Analysis: Inference
11.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Wage equation (cont.) âThe effect of experience on hourly wage is statistically greater than zero at the 5% (and even at the 1%) significance level.â t-statistic Degrees of freedom; here the standard normal approximation applies Critical values for the 5% and the 1% significance level (these are conventional significance levels). The null hypothesis is rejected because the t-statistic exceeds the critical value. Multiple Regression Analysis: Inference
12.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Test against . Reject the null hypothesis in favour of the alternative hypothesis if the estimated coef- ficient is âtoo smallâ (i.e. smaller than a criti- cal value). Construct the critical value so that, if the null hypothesis is true, it is rejected in, for example, 5% of the cases. In the given example, this is the point of the t- distribution with 18 degrees of freedom so that 5% of the cases are below the point. ! Reject if t-statistic less than -1.734 Multiple Regression Analysis: Inference Testing against one-sided alternatives (less than zero)
13.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Student performance and school size Test whether smaller school size leads to better student performance Test against . Do larger schools hamper student performance or is there no such effect? Percentage of students passing maths test Average annual tea- cher compensation Staff per one thou- sand students School enrollment (= school size) Multiple Regression Analysis: Inference
14.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Student performance and school size (cont.) One cannot reject the hypothesis that there is no effect of school size on student performance (not even for a lax significance level of 15%). t-statistic Degrees of freedom; here the standard normal approximation applies Critical values for the 5% and the 15% significance level. The null hypothesis is not rejected because the t-statistic is not smaller than the critical value. Multiple Regression Analysis: Inference
15.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Student performance and school size (cont.) Alternative specification of functional form: Test against . R-squared slightly higher Multiple Regression Analysis: Inference
16.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Student performance and school size (cont.) The hypothesis that there is no effect of school size on student performance can be rejected in favor of the hypothesis that the effect is negative. t-statistic Critical value for the 5% significance level ! reject null hypothesis How large is the effect? + 10% enrollment ! -0.129 percentage points students pass test (small effect) Multiple Regression Analysis: Inference
17.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing against two-sided alternatives Test against . Reject the null hypothesis in favour of the alternative hypothesis if the absolute value of the estimated coefficient is too large. Construct the critical value so that, if the null hypothesis is true, it is rejected in, for example, 5% of the cases. In the given example, these are the points of the t-distribution so that 5% of the cases lie in the two tails. ! Reject if absolute value of t-statistic is less than -2.06 or greater than 2.06 Multiple Regression Analysis: Inference
18.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Determinants of college GPA Lectures missed per week The effects of hsGPA and skipped are significantly different from zero at the 1% significance level. The effect of ACT is not significantly different from zero, not even at the 10% significance level. For critical values, use standard normal distribution Multiple Regression Analysis: Inference
19.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. âStatistically significantâ variables in a regression If a regression coefficient is different from zero in a two-sided test, the corresponding variable is said to be âstatistically significantâ If the number of degrees of freedom is large enough so that the nor- mal approximation applies, the following rules of thumb apply: âstatistically significant at 10 % levelâ âstatistically significant at 5 % levelâ âstatistically significant at 1 % levelâ Multiple Regression Analysis: Inference
20.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Guidelines for discussing economic and statistical significance If a variable is statistically significant, discuss the magnitude of the coefficient to get an idea of its economic or practical importance The fact that a coefficient is statistically significant does not necessa- rily mean it is economically or practically significant! If a variable is statistically and economically important but has the âwrongâ sign, the regression model might be misspecified If a variable is statistically insignificant at the usual levels (10%, 5%, 1%), one may think of dropping it from the regression If the sample size is small, effects might be imprecisely estimated so that the case for dropping insignificant variables is less strong Multiple Regression Analysis: Inference
21.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing more general hypotheses about a regression coefficient Null hypothesis t-statistic The test works exactly as before, except that the hypothesized value is substracted from the estimate when forming the statistic Hypothesized value of the coefficient Multiple Regression Analysis: Inference
22.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Campus crime and enrollment An interesting hypothesis is whether crime increases by one percent if enrollment is increased by one percent The hypothesis is rejected at the 5% level Estimate is different from one but is this difference statistically significant? Multiple Regression Analysis: Inference
23.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Computing p-values for t-tests If the significance level is made smaller and smaller, there will be a point where the null hypothesis cannot be rejected anymore The reason is that, by lowering the significance level, one wants to avoid more and more to make the error of rejecting a correct H0 The smallest significance level at which the null hypothesis is still rejected, is called the p-value of the hypothesis test A small p-value is evidence against the null hypothesis because one would reject the null hypothesis even at small significance levels A large p-value is evidence in favor of the null hypothesis P-values are more informative than tests at fixed significance levels Multiple Regression Analysis: Inference
24.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. How the p-value is computed (here: two-sided test) The p-value is the significance level at which one is indifferent between rejecting and not rejecting the null hypothesis. In the two-sided case, the p-value is thus the probability that the t-distributed variable takes on a larger absolute value than the realized value of the test statistic, e.g.: From this, it is clear that a null hypothesis is rejected if and only if the corresponding p- value is smaller than the significance level. For example, for a significance level of 5% the t-statistic would not lie in the rejection region. value of test statistic These would be the critical values for a 5% significance level Multiple Regression Analysis: Inference
25.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Critical value of two-sided test Confidence intervals Simple manipulation of the result in Theorem 4.2 implies that Interpretation of the confidence interval The bounds of the interval are random In repeated samples, the interval that is constructed in the above way will cover the population regression coefficient in 95% of the cases Lower bound of the Confidence interval Upper bound of the Confidence interval Confidence level Multiple Regression Analysis: Inference
26.
© 2013 Cengage
Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence intervals for typical confidence levels Relationship between confidence intervals and hypotheses tests reject in favor of Use rules of thumb Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Model of firmsâ R&D expenditures Spending on R&D Annual sales Profits as percentage of sales The effect of sales on R&D is relatively precisely estimated as the interval is narrow. Moreover, the effect is significantly different from zero because zero is outside the interval. This effect is imprecisely estimated as the in- terval is very wide. It is not even statistically significant because zero lies in the interval. Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing hypotheses about a linear combination of parameters Example: Return to education at 2 year vs. at 4 year colleges Years of education at 2 year colleges Years of education at 4 year colleges Test against . A possible test statistic would be: The difference between the estimates is normalized by the estimated standard deviation of the difference. The null hypothesis would have to be rejected if the statistic is âtoo negativeâ to believe that the true difference between the parameters is equal to zero. Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Insert into original regression Impossible to compute with standard regression output because Alternative method Usually not available in regression output Define and test against . a new regressor (= total years of college) Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Estimation results This method works always for single linear hypotheses Total years of college Hypothesis is rejected at 10% level but not at 5% level Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing multiple linear restrictions: The F-test Testing exclusion restrictions Years in the league Average number of games per year Salary of major lea- gue base ball player Batting average Home runs per year Runs batted in per year against Test whether performance measures have no effect/can be exluded from regression. Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Estimation of the unrestricted model None of these variabels is statistically significant when tested individually Idea: How would the model fit be if these variables were dropped from the regression? Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Estimation of the restricted model Test statistic The sum of squared residuals necessarily increases, but is the increase statistically significant? The relative increase of the sum of squared residuals when going from H1 to H0 follows a F-distribution (if the null hypothesis H0 is correct) Number of restrictions Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Rejection rule (Figure 4.7) A F-distributed variable only takes on positive values. This corresponds to the fact that the sum of squared residuals can only increase if one moves from H1 to H0. Choose the critical value so that the null hypo- thesis is rejected in, for example, 5% of the cases, although it is true. Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Test decision in example Discussion The three variables are âjointly significantâ They were not significant when tested individually The likely reason is multicollinearity between them Number of restrictions to be tested Degrees of freedom in the unrestricted model The null hypothesis is overwhel- mingly rejected (even at very small significance levels). Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Test of overall significance of a regression The test of overall significance is reported in most regression packages; the null hypothesis is usually overwhelmingly rejected The null hypothesis states that the explanatory variables are not useful at all in explaining the dependent variable Restricted model (regression on constant) Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing general linear restrictions with the F-test Example: Test whether house price assessments are rational The assessed housing value (before the house was sold) Size of lot (in feet) Actual house price Square footage Number of bedrooms If house price assessments are rational, a 1% change in the assessment should be associated with a 1% change in price. In addition, other known factors should not influence the price once the assessed value has been controlled for. Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Unrestricted regression Restricted regression Test statistic The restricted model is actually a regression of [y-x1] on a constant cannot be rejected Multiple Regression Analysis: Inference
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Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Regression output for the unrestricted regression The F-test works for general multiple linear hypotheses For all tests and confidence intervals, validity of assumptions MLR.1 â MLR.6 has been assumed. Tests may be invalid otherwise. When tested individually, there is also no evidence against the rationality of house price assessments Multiple Regression Analysis: Inference
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