ECON 7310: ELEMENTS OF ECONOMETRICS Dr. Dong-Hyuk Kim Tutorial 4: Multiple Linear Regression, Part A At the end of this tutorial you should be able to • understand and explain the assumptions behind the multiple regression model; • explain the properties of the least squares estimators; • compute and interpret the least squares regression coefficients; • estimate the variances and covariances of the least squares estimators; • conduct hypothesis tests and construct confidence intervals concerning the regression coefficients; • use the estimated regression model to generate predictions; and • compute and explain measures of goodness-of-fit. Problems: 1. (Based on PE4 Ex.5.24) The file rice.dta contains 352 observations on 44 rice farmers in the Tarlac region of the Phillipines for the 8 years 1990 to 1997. Variables in the data set are tonnes of freshly threshed rice (prod), hectares planted (area), person-days of hired and family labour (labor) and kilograms of fertiliser (fert).Treat the data set as one sample with N = 352 and answer the following questions. Estimate the production function log(prod) = β1 + β2 log(area) + β3 log(labor) + β4 log(fert) + e (a) Report the results and interpret the estimates. What proportion of the variation in production is explained by the estimated regression model. (Answer) First, we need to define new variables Then, we run the regression as follows; 1All estimates have elasticity interpretations. For example, a 1% increase in labour will lead to a 0.4328% increase in rice output. A 1% increase in fertiliser will lead to a 0.2095% increase in rice output. Note that this is log − log linear model. Hence, prod [ = exp log( \ prod) = exp (b1 + b2 log(area) + b3 log(labor) + b4 log(fert)) where (b1; b2; b3; b4) are the OLS estimates. Then, the generalised R2 = 0:9122 = 0:8318.  (b) Comment on the statistical significance of the individual coefficient estimates. (Answer) We say that the coefficient estimate bk is significant if we reject the hypothesis H0 : βk = 0. For this hypothesis testing, the decision rule is to reject H0 if the test statistics bk − 0 se(bk) is larger than the critical value, which depends on the level α of test. The t-value in the third column of computer outcome represents this ratio. All of them are larger than the commonly used critical values (e.g., 1.96 when α = 0:05). Hence, all the estimates are statistically significant. Alternatively, we could use the p values for testing a hypothesis. Decision rule is simple; we reject H0 : β0 = 0 if p-value is less than α. The fourth column of the output shows the p values, which 2are all essentially zero. Hence, for any conventionally used level (e.g., α = 0:05 or 0.01), the estimates are all statistically significant.  (c) Using a 5% level of significance, test the hypothesis that the elasticity of production with respect to land is equal to 0.5. (Answer) We test H0 : β2 = 0:5 against H1 : β2 6= 0:5 at the level of α = 0:05. We reject H0 because b2 − 0:5 se(b2) = 0:3617359 − 0:5 0:0639678 = 2:1615 > 1:96  (d) Find a 95% interval estimate for the elasticity of production with respect to fertiliser. Has this elasticity been precisely measured? (Answer) Even before we construct the confidence interval, we know the estimate is precise because it is statistically significant. The confidence interval is given as b4 ± 1:96 × se(b4) = 0:2095 ± 1:96 × 0:0382654 = [0:1345; 0:2845]:  (e) Test the significance of the regression model. Use α = 0:05 (Answer) We test H0 : β2 = β3 = β4 = 0 against H1 : H0 is wrong, i.e., β2 6= 0 or β3 6= 0 or β4 6= 0. The Stata outcome shows the F statistic for this testing, F = 646:51. Moreover, the associated p-value is zero. So, we reject H0.  (f) The production technology exhibits constant returns to scale if β2 + β3 + β4 = 1. Test the null hypothesis of constant returns to scale using a 5% level of significance. (Answer) After running the regression in part (a), click on Statistics ! Postestimation ! Tests ! Test linear hypotheses. Then, you will Click on Create and choose Linear expressions are equal in Test type: 3In the Coefficient: section, you can add coefficients. Add the coefficients associated with log area, log labor, and log fert. Then, since the defaulted relationship between coefficients is =, you will have Edit the Linear expression: as follows; 4Then, click on Ok and Submit. Then, the result will show up as follows; F-statistic is 0.03, which is very small. Moreover, the p-value is 0:8638 > α = 0:05. So, we do not reject H0. The hypothesis of constant returns to scale cannot be rejected at the 5% significance level.  (g) Predict rice production from 1 hectare of land, 50 person-days of labour and 100 kg of fertiliser. (Answer) log( \ prod) = −1:5468 + 0:3617 log(1) + 0:4328 log(50) + 0:2095 log(100) = 1:111107 Therefore, prod [ = exp(1:111107) = 3:0377.  (h) Your economic principles suggest that a farmer should continue to apply fertiliser as long as the marginal product of fertiliser @prod=@fert is greater than the price of fertiliser divided by the price of output. Suppose that this price ratio is 0.004. For fert = 100 and the predicted value for prod found in part g), show that the farmer should continue to apply fertiliser as long as β4 > 0:1242: (Answer) The model can be written as prod = exp (β1 + β2 log(area) + β3 log(labor) + β4 log(fert) + e) Then, @prod @fert = @ @fert exp (β1 + β2 log(area) + β3 log(labor) + β4 log(fert) + e) = exp (β1 + β2 log(area) + β3 log(labor) + β4 log(fert) + e) × @fert @ β4 log(fert) = exp (log(prod)) × β4 fert 1  = β4 × prod fert: Now, the marginal principle says we should apply more fertiliser if @prod @fert > Price of fert Price of prod = 0:004: Since @prod @fert = β4 × prod fert; we should apply more fertiliser if β4 × prod fert > 0:004 5When prod = 3:0377 and fert = 100, this condition becomes β4 > 0:1317  (i) Using H1 : β4 > 0:1317 as the alternative hypothesis, test whether the farmer should apply more fertiliser. Use a 5% level of significance. Why did we choose as the alternative hypothesis? (Answer) We test H0 : β4 = 0:1317 against H1 : β4 > 0:1317. For this one-sided test, we would reject H0 when the estimate b4 is too large. More specifically, we reject H0 when b4 − 0:1317 se(b4) = 0:2095 − 0:1317 0:0383 = 2:03 is larger than the critical value of the test with level of α = 0:05. From the probability table of the standard normal distribution, we can find that Pr(Z < 1:645) = 0:95, So, the critical value is 1.645, which is smaller than the test statistic 2.03. Therefore, we reject H0. We conclude that at the 5% significance level, the farmer should apply more fertiliser. We chose a one-sided alternative hypothesis following the marginal principle in part (h).  (j) Find a 95% interval estimate for rice production from 1 hectare of land, 50 person-days of labour and 1 kg of fertiliser. (Answer) Recall that log(1) = 0. We can compute log(50) by typing display log(50) in the Command window. The Stata would give you log(50) = 3:912023. Now, re-run the regression in part (a) and run adjust command as shown below; We see that the predicted value of log(prod) = 0:146525 with prediction standard error of se(f) = 0:37938, which gives the 95% prediction interval of [−0:599641; 0:892691]. Since this is the interval estimate for log(prod), the interval estimate for prod is [exp(−0:599641); exp(0:892691)] = [0:5490; 2:4417]. (The Stata command adjust must follow an estimation command.) 6