ECA5103 Assignment 2 All students have to hand in this assignment by 11:59pm Friday 29 October 2021. Please submit both your answers and program. 1. Apartment developers sometimes add extra greenery, like more trees, to the premises of an apartment project, hoping that it leads to higher apartment prices. Suppose you estimate a simple linear regression with dependent variable ln price, log price of an apartment, and the lone explanatory variable is greenery, which equals 1 if the project to which the apartment belongs has extra greenery and equals 0 otherwise: ln pricei = γ0 + γ1greeneryi + ui, i = 1, . . . , n (a) Suggest an omitted variable that would cause the OLS estimator γˆ1 to be inconsistent for β1, the true effect of greenery on ln price. (b) Use the omitted variable bias formula to sign the direction of the bias: plim γˆ1 = β1 + β2 cov (x1, x2) var (x1) (c) Suppose that you didn’t realize that omitted variable bias was a problem. How would the bias that you find in part (b) affect your conclusion regarding the effect of greenery on apartment prices? 2. Assume that the relationship between unemployment rate u and inflation rate i is determined by the following equation ut = 0.3it + 0.1it−1 − 0.02it−2 (1) where ut is the unemployment rate in year t, it is the inflation rate in year t, and i ∼ iidN(1, 4). (a) Compute the mean and variance of ut. 1 ECA5103 Assignment 2 (b) Compute the first three autocovariances of ut: cov (ut, ut−1) , cov (ut, ut−2) , cov (ut, ut−3). (c) Compute the first three autocorrelations of ut. (d) Is ut stationary? Is it weakly dependent? 3. The file beer contains monthly beer sales for a manufacturer over a number of years. (a) The data includes a year and a month variable. Let’s combine this into a single variable that we’ll call period. The command is ym. After creating period, format the variable using the command format period %tm. Use the tsset to tell Stata that the relevant time variable is period. (b) Create two new variables, log of beer sales and quarter of the year, which we will use throughout the analysis. (c) Graph the log of beer sales over time. Comment on the pattern. (d) Estimate a regression that accounts for seasonality (at the quarterly level) and statisti- cally test if there is seasonality in the data. Comment. (e) Compute the residuals from the regression and plot them against time. Comment on the pattern. (f) Estimate an AR(1) of the residuals. Do the results indicate that the residuals are serially correlated? (g) Re-estimate the model now adding the lag of log beer sales as an additional right-hand side variable. Comment on the results. (h) Re-do parts (e) and (f) using the modified model. What do you conclude? Which model do you prefer? (i) Estimate an ARCH(1) model. Comment on your findings. (Control seasonal variation at quarterly level) 2