OBS 1975Q2 1975Q3 1975Q4 1976Q1 1976Q2 1976Q3 1976Q4 1977Q1 1977Q2 1977Q3 1977Q4 1978Q1 1978Q2 1978Q3 1978Q4 1979Q1 1979Q2 1979Q3 1979Q4 1980Q1 1980Q2 1980Q3 1980Q4 1981Q1 1981Q2 1981Q3 1981Q4 1982Q1 1982Q2 1982Q3 1982Q4 1983Q1 1983Q2 1983Q3 1983Q4 1984Q1 1984Q2 1984Q3 1984Q4 1985Q1 1985Q2 1985Q3 1985Q4 1986Q1 1986Q2 1986Q3 1986Q4 1987Q1 1987Q2 1987Q3 1987Q4 1988Q1 G 3.875226 2.784185 3.231245 3.026323 0.800703 3.828554 4.758848 8.321665 6.206399 8.27685 5.84781 4.149952 4.963949 4.376981 6.023289 3.121601 6.391586 3.740141 2.591726 1.817442 1.787449 -2.848096 -10.60027 17.50684 4.861545 -1.977469 -7.039502 4.628838 2.53782 -2.605512 -0.360535 1.718688 0.523817 0.430599 1.50887 0.454257 1.057327 1.195765 0.272555 1.564087 1.481622 1.86904 -1.016139 2.910147 2.000959 2.357219 1.48631 2.107113 2.519729 2.445134 2.276081 2.795112 1988Q2 1988Q3 1988Q4 1989Q1 1989Q2 1989Q3 1989Q4 1990Q1 1990Q2 1990Q3 1990Q4 1991Q1 1991Q2 1991Q3 1991Q4 1992Q1 1992Q2 1992Q3 1992Q4 1993Q1 1993Q2 1993Q3 1993Q4 1994Q1 1994Q2 1994Q3 1994Q4 1995Q1 1995Q2 1995Q3 1995Q4 1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 2000Q1 2000Q2 2000Q3 2000Q4 2001Q1 2001Q2 3.945252 4.448373 6.066199 5.131782 5.39785 3.541867 2.642109 0.765248 0.32053 0.622361 -2.151717 -0.90379 -0.534289 -0.340676 0.290253 -0.209403 -1.283202 -0.410903 -2.039262 -2.583085 -0.58374 -1.634627 -1.204939 0.340854 0.331579 -0.570511 -0.929151 -0.862577 -0.300372 -1.540487 -0.468912 0.915116 0.755921 0.198175 -1.417731 1.997965 2.330914 1.592611 1.507377 3.249282 5.516339 1.711208 0.663682 3.711203 3.768753 2.375707 3.234623 3.432154 5.126509 2.849223 2.672024 3.45 3.348252 2001Q3 2001Q4 2002Q1 2002Q2 2002Q3 2002Q4 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4 2005Q1 2005Q2 2005Q3 2005Q4 2006Q1 2006Q2 2006Q3 2006Q4 2007Q1 2007Q2 2007Q3 2007Q4 2008Q1 2008Q2 2008Q3 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 2010Q1 2010Q2 2010Q3 2010Q4 2011Q1 2011Q2 2011Q3 2011Q4 2012Q1 2012Q2 3.080927 1.422937 5.092976 7.109088 5.802894 3.151133 3.535192 4.303659 5.612785 3.59723 5.843555 9.546292 3.625518 2.159167 0.516565 3.805385 2.019955 -1.031314 -0.348242 -0.340119 -3.491283 -1.844434 0.618136 -1.169765 -4.9801 -7.424729 -6.934659 -3.430527 -5.040843 -6.619759 -3.595453 3.990836 3.516095 1.2723 0.926239 0.261861 -1.794278 -1.91266 -3.081349 0.686884 -2.400453 -0.954674 1.326932 1.328486 ECON 504 Spring 2017 HW 7 (Due in class on March 29, 2017) 1. State for each of the following time series processes: whether it is stationary, and whether it is invertible. Assume Wt~i.i.d.N(0,1). a. Zt = 3.3 + Wt -1.3 Wt-1 + 0.3Wt-2 b. Xt = 0.5 Xt-1 + Wt c. Xt - Xt-1 + 0.5Xt-2 = 0.5 Wt-1 + Wt d. Xt - Xt-1 = Wt + 5Wt-1 e. Yt = 0.5Yt-1 + 0.4Yt-2 - 0.5 Yt-3 + 0.4 Yt-4 + 0.5 Yt-5 + 0.3 Yt-6 + Wt f. Zt = 9 + Wt 2. For this exercise, use the annual data on per capita income (nominal) in the state of New York from http://bea.gov/regional/index.htm. Use the data for the period 1950-2015 for your analysis. a. Identify and estimate an ARMA model for per capita income growth in New York. Use R. Explain the rationale behind the choice of your model. Include at least one other potential candidate model for your comparison. b. Use R to obtain and present point forecasts, and the standard errors of the forecasts for years 2016 and 2017 based on your chosen model. Ignore the uncertainty arising from the estimation error for your calculation of the standard error of your forecasts. What are the corresponding density forecasts? c. Use Excel to derive and present the quantiles of your forecast density for 2016 in the same way as they are presented in slide # 30 of lecture note#14:Identification and Estimation of ARMA. 3. Use the San Diego house price data (data on the growth on the housing prices: G) that I have posted with this homework. Use the data up to 2008:2 as the estimation sample and the data in 2008:3 - 2012:2 as the prediction sample. a. Estimate an AR(4) model of the growth of the housing price in San Diego. Then using a fixed scheme, make one-period-ahead forecasts based on the model. Call the forecast FA. The purpose of the exercise is forecast evaluation. b. Generate a second series of one-period-ahead forecasts that are calculated by averaging the last two observations. Call it FB. c. Present a table similar to Table 9.2 in the text book of the two sets of forecasts and forecast error series along with the actual data. d. Implement the forecast optimality tests (viz., MPE test, and Information Efficiency test assuming quadratic loss function) on your two forecasts. Present your results. (That is, present tables similar to the ones in Table 9.4 in the text book - excluding the Linex loss function.) e. Implement a descriptive evaluation of the average loss resulting from the three forecasts. Use RMSE, MAE and MAPE as the measures of the average loss. Summarize your results in a table like Table 9.6 (excluding Linex loss function) in the text book. f. Now statistically test whether the squared error loss from FB is any different from the loss from FA. What is your conclusion