Empirical Application of Simple Regression (Chapter 2)

Transcription

1 Empirical Application of Simple Regression (Chapter 2) 1. The data file is House Data, which can be downloaded from my webpage. 2. Use stata menu File Import Excel Spreadsheet to read the data. Don t forget to check the box that import the first row as variable names. 3. You can view the data using command bro. You can summarize variables using command sum. 4. How to select the y variable depends on purpose and context. We use econ theory or common sense to find the x variable. 5. Here, the dependent variable is y = rprice, the real house price in 1978 dollars (using 1978 consumer price index, or GDP deflator). We want to explain or forecast the real house price. 6. Simple regression uses just one independent variable (regressor) to explain y. Here x = baths, the number of bathrooms in a house. Common sense tells us the number of bathroom should matter for the house price. (a) To motivate the simple regression, consider the command to get a scatter plot twoway (scatter rprice baths) rprice baths (b) It seems that the average house price rises as baths rises. In other words, the population regression function can be represented by an upward-sloping line. 1

2 (c) Or you can use these commands to check the pattern in the conditional mean: sort baths by baths: sum rprice Critical thinking: what is the difference between unconditional and conditional means? Which one is more useful? (d) All other factors that affect the house price is captured by the error term u, and that turns out to be the biggest limitation of the simple regression. Basically the simple regression ignores all other factors. (e) For example, u includes area, the square footage of a house. We do have data for this variable, but simple regression effectively treats it as unobserved. (f) The simple regression is rprice = β 0 + β 1 baths + u. (g) β 0 is the intercept. Mathematically, β 0 = E(rprice baths = 0). In words, the intercept measures the average house price for houses with zero bathroom. In many cases like this one, the intercept is not very meaningful. (h) If we assume cov(x, u) = 0, it follows that y = β 1 x, or y = β 1 when x = 1. So if baths is uncorrelated with all other factors (a very strong assumption), then β 1 measures the change in house price when one more bathroom is added to the house. In that case, β 1 has causal interpretation. If cov(x, u) 0, β 1 has no causal interpretation, and it only measures the association or correlation. Critical thinking: do you think cov(x, u) = 0 if x is baths and u is area? (i) A common question asked by a home seller is is it a good idea to add a new bathroom to an existing house so that it will be easier to sell the house? From the economics viewpoint, the answer is it makes economic sense as long as the cost the new bathroom is less than β 1, the marginal value of adding a new bathroom. (j) In this example, because u includes area, and common sense tells us a big house tends to have more bathrooms than a small house. So we have cov(baths, area) 0 No causal interpretation of β 1 So the simple regression cannot be used to make the decision of adding vs not 2

3 adding a new bathroom, simply because β 1 does not have causal interpretation here. (k) Since the assumption cov(x, u) 0 fails here, β 1 only measures the association (or correlation). We can only say adding one bathroom is associated with the change of house price by β The stata command reg rprice baths reports the OLS estimate of β 1 and β 0.. reg rprice baths Source SS df MS Number of obs = F( 1, 319) = Model e e+11 Prob > F = Residual e R-squared = Adj R-squared = Total e e+09 Root MSE = rprice Coef. Std. Err. t P> t [95% Conf. Interval] baths _cons We have ˆβ 1 = ; ˆβ 0 = Exercise (a) How to obtain ˆβ 1 using command corr rprice baths, cov (b) How to interpret ˆβ 1? Why does it have no causal interpretation? (c) How to obtain ˆβ 0 using command sum rprice baths, after obtaining ˆβ 1? 3

4 (d) What will happen to ˆβ 1 if we divide baths by 2? Does this mean you can manipulate ˆβ 1? 9. We can prove that the OLS estimator has some nice properties such as unbiasedness. In particular under the additional assumption of homoskedasticity var(u x) = σ 2 we have var( ˆβ 1 ) = se( ˆβ 1 ) = σ 2 (xi x) 2 (1) σ (xi x) 2 (2) t value = ˆβ 1 β 1 se( ˆβ 1 ) N(0, 1) (3) In practice, σ 2 is unknown, and is estimated by where û is the residual: ˆσ 2 = 1 û2 n 2 i, û i = y i ŷ i = rprice i ˆβ 0 ˆβ 1 baths i = rprice i baths i. 10. Exercise (a) Compute the first and second residuals. How to interpret them? Hint: the fitted value and residual can be obtained using the command predict yhat predict uhat, r (b) Compute ˆσ, which is called the standard error of regression (SER) or root mse (stata terminology). Hint: gen u2 = uhat^2 qui sum u2 return list 4

5 sca rootmse = sqrt(r(sum)/(321-2)) dis "root mse is " rootmse (c) Compute the standard error of ˆβ 1. Hint: (x i x) 2 = (n 1)Sx 2 qui sum baths sca se = rootmse/sqrt((321-1)*r(var)) dis "standard error is " se (d) Compute the t value of ˆβ 1 using the (default) null hypothesis H 0 : β 1 = 0. How to interpret this null hypothesis? What is your conclusion? (e) What will happen to t value of ˆβ 1 if we divide baths by 2? 5