Econ 493 A1 – Fall 2019 Homework 4 Assignment Information This assignment is due on Monday November 18 at 11:59 am. Submit the assignment in the locked box in the Department of Economics General Office (Tory 8-14). Note that the General Office is CLOSED daily from 12-1 pm and after 4:00 pm. Late assignments will receive NO MARKS. Answers to computing exercises must include R commands and output files when applicable. All answers must be transcribed to your written answers which must be separate from the R printout. Total marks = 50 (5 questions). Exercise 1 Electricity consumption is often modelled as a function of temperature. Temperature is measured by daily heating degrees and cooling degrees. Heating degrees is 18◦C minus the average daily temperature when the daily average is below 18◦C; otherwise it is zero. This provides a measure of our need to heat ourselves as temperature falls. Cooling degrees measures our need to cool ourselves as the temperature rises. It is defined as the average daily temperature minus 18◦C when the daily average is above 18◦C; otherwise it is zero. Let yt denote the monthly total of kilowatt-hours of electricity used, let x1,t denote the monthly total of heating degrees, and let x2,t denote the monthly total of cooling degrees. An analyst fits the following model to a set of such data: y∗t = β1x∗1,t + β2x∗2,t + ηt, where (1−B)(1−B12)ηt = 1− θ1B1− φ12B12 − φ24B24 εt and y∗t = log(yt), x∗1,t = √ x1,t and x∗2,t = √ x2,t. a. What sort of ARIMA model is identified for ηt? b. The estimated coefficients of β1 and β2 are found to be greater than zero. Explain what the estimates of β1 and β2 tell us about electricity consumption. c. Describe how this model could be used to forecast electricity demand for the next 12 months. d. Explain why the ηt term should be modelled with an ARIMA model rather than modeling the data using a standard regression package. In your discussion, comment on the properties of the estimates, the validity of the standard regression results, and the importance of the ηt model in producing forecasts. 1 Exercise 2 Given an initial value for y0, re-write each yt in terms of y0 and past innovations (that is, εi for i = 0, . . . , t). Also, find the h-step-ahead forecast for h = 1, 2. a. yt = yt−1 + εt + 0.5εt−1 b. yt = 1.1yt−1 + εt c. yt = yt−1 + 1 + εt d. yt = yt−1 + t+ εt Exercise 3 (R) The file us_macro_quarterly.csv contains quarterly data on several macroeconomic series for the United States. The variable PCEP is the price index for personal consumption expenditures from the US National Income and Product Accounts. In this exercise you will construct forecasting models for the rate of inflation, based on PCEP . For this analysis, use the sample period 1963Q1 to 2012Q4. a. Compute the inflation rate, inflt = 400 × [log(PCEPt) − log(PCEPt−1)]. What are the units of infl? b. Use R to plot the inflation rate series (infl) and the ACF. Does the series appear to be stationary? Explain. c. Use R to plot the change in the inflation rate series (infl′) and the ACF. Does the differenced series appear to be stationary? Explain. d. Use the ADF test to determine d. e. Compute and plot the one-step-ahead quarterly forecasts of the inflation rate for the pseudo out-of-sample period 2003Q1 to 2012Q4 (40 quarters) using the following models: (i) an ARIMA(2,0,0) and (ii) and ARIMA(2,1,0). Compare your results in terms of the RMSE. f. Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a non-zero mean? Exercise 4 (R) Consider the spurious regression problem with time series data. The file inflation.csv contains 39 annual observations of the following variables (by columns): – Year: 1971-2009 – Deaths: Total number of deaths, Canada – CPI: Consumer Price Index, Canada a. Use the CPI series to compute the annual inflation rate, inflt = 100 × [log(CPIt) − log(CPIt−1)] for the sample 1972–2009. Plot the time series. b. Obtain the total number of deaths in Canada per 1000 people for the sample 1972–2009 (that is, divide the data by 1000). Plot the time series. c. Use OLS to estimate the equation inflt = β0 + β1deadt + εt. Is deaths significant at the 5% level? What is the sign of the slope coefficient? d. Relate your results in (c) to the spurious regression problem. 2 e. Use OLS to estimate the equation infl′t = β0 + β1dead′t + εt. Is deaths significant at the 5% level? What is the sign of the slope coefficient? f. Use OLS to estimate the equation inflt = β0+β1deadt+β2time+εt. Is deaths significant at the 5% level? What is the sign of the slope coefficient? Exercise 5 (R) The file quarterly.csv contains the series of quarterly industrial production and the con- sumer’s price index (CPI) for the US for the quarters 1960:Q1 to 2012:Q4. a. Create the log change in the index of industrial production (indprod) as lip′ = log(indprodt) − log(indprodt−1) and the inflation rate as inflt = log(CPIt) − log(CPIt−1). b. Determine if lip′ and inflt are stationary? c. Estimate the bivariate VAR using three lags of each variable and a constant. Verify that the three-lag specification is selected by the BIC, whereas the AIC selects five lags. d. Perform the Granger causality tests. Verify that the F-statistic for the test that inflation Granger-causes industrial production is 4.82 (with a significance level of 0.003) and that the F-statistic for the test that industrial production Granger-causes inflation is 5.1050 (withh a significance level of 0.002). 3
