Regression Analysis Tutorial 109Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999LECTURE / DISCUSSIONHypothesis TestingRegression Analysis Tutorial 110Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999IntroductionConsider the model you estimated in the previousworkshop. The output for the first model is: Dependent variable: USAGE Current sample: 1 to 500 Number of observations: 500 Mean of dependent variable = 229970. Std. dev. of dependent var. = 91538.8 Sum of squared residuals = .362272E+13 Variance of residuals = .731863E+10 Std. error of regression = 85549.0 R-squared = .133589 Adjusted R-squared = .126588 Durbin-Watson statistic = 2.06459 F-statistic (zero slopes) = 19.0806 Schwarz Bayes. Info. Crit. = 22.7658 Log of likelihood function = -6385.38 Estimated Standard Variable Coefficient Error t-statistic C 127568. 15133.6 8.42949 AC -20606.7 8116.66 -2.53882 CDD 23.9860 4.80534 4.99152 NEMPLOY -6.61867 89.4999 -.073952 SQFT 931.616 147.957 6.29652The point estimates are in the first column of numbers. Thislecture introduces you to the use of the other columns.Regression Analysis Tutorial 111Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999The second column gives numbers that describe thestatistical precision of the point estimates. Because theregression relationship includes a disturbance term, onecannot estimate the coefficients on the explanatory variablesexactly. If a new data set were collected, and the sameregression coefficients re-estimated, the new estimateswould not equal the initial ones because the OLS estimatorsare random variables. The second column containsestimates of the standard deviations of each OLS regressioncoefficient; these estimates are commonly called standarderrors. A common rule of thumb is that the actualregression coefficient is probably within two standarderrors of its point estimate. Thus, the actual audit impact isprobably between 4,375. and 36,839.Regression Analysis Tutorial 112Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999The third column of numbers, labeled t-statistics, gives theratio of the point estimates over their standard errors. Theseratios describe the likelihood that the actual coefficient iszero. Using the ‘‘two standard errors’’ rule of thumb, anactual coefficient is unlikely to be zero if the t-statistic islarger than 2 in absolute value. In these estimates, it is likelythat the number of employees has no impact on monthlyelectricity consumption in a building (given its size and theweather).Regression Analysis Tutorial 113Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999ˆy 'jKk'1ˆkxkIn the material that follows, we explain the basis for the‘‘two standard errors’’ rule and how this rule is generalizedto examining several coefficients simultaneously. If we aregoing to forecast energy consumption with this estimatedregression, we will want to compute standard errors for theforecasts that include the variation in a combination of theestimated coefficients:is a forecast of y given the explanatory variable values x. Another way we might be concerned with manycoefficients simultaneously occurs in combining distinctdata sets: before pooling all observations in a single, larger,data set, we want to check whether there is any evidencethat the actual regression coefficients are different in thedistinct data sets. This is analogous to asking whether thereis evidence for a set of coefficients where the coefficients arenot zero.Regression Analysis Tutorial 114Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999ˆk&ksk- tN& KConfidence IntervalsIf the regression model is specified correctly, so that noexplanatory variables are missing, thenwhere $k is the true value of the kth explanatory variable, is the OLS estimator of this coefficient, and sk is theˆkestimated standard error of . The symbol tN-Kˆkrepresents a random variable with a particular distribution:Student’s t with N - K degrees of freedom. We will usethis fact to:1. Construct confidence intervals for individualregression coefficients.2. Test hypotheses about the individual regressioncoefficients.Regression Analysis Tutorial 115Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999Student’s t DistributionThe Student’s t distribution, or simply t distribution, is ageneralization of the standard normal distribution. Thedistribution is characterized by an additional parameter,called the degrees of freedom. For small degrees of freedom(less than 10), the t distribution has a probability densityfunction that is much ’fatter’ than the normal distribution:the tails of the density approach zero more slowly. As thedegrees of freedom increase, the t distribution approachesthe normal distribution so that in the limit the twodistributions coincide. When the degrees of freedom exceed30, the coincidence is quite close.Regression Analysis Tutorial 116Econometrics Laboratory C University of California at Berkeley C 22-26 March 199900.050.10.150.20.250.30.350.4-4-3 -2 -1 0 1 2 34.................................................................................Values of the Random VariableProbability Density Functiont Distribution (2 d.f.)Normal DistributionStudent’s t and Normal Probability Density FunctionsThe statistic has been standardized by(ˆk&k)/sksubtracting from its expectation and then dividing theˆkresult by the estimated standard error. Given the correctvalue of $k , this statistic has a t distribution with degreesof freedom equal to the number of observations in thesample minus the total number of explanatory variables inthe estimated regression. The distribution of /sk(ˆk&k)does not depend on any unknown parameters.Regression Analysis Tutorial 117Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999Pr a #ˆk&ksk# b ' pProbability IntervalsBecause we can compute the probability density function ofany t distribution, given any interval [a,b], we cancompute the probability that will fall in the(ˆk&k)/skinterval if we are given a brand new data set of Nobservations. Given a probability p