Page 2 of 9 PART A: MULTIPLE-CHOICE QUESTIONS (EACH QUESTION IS WORTH 1 POINT) 1. A random walk is a stationary process for which all autocorrelations equal to zero. A. TRUE B. FALSE 2. Consider a simple Cobweb model, where the demand is given by, = − , and the supply is given by = + ∗ + , where ∗ is the expected price, and where is an iid white noise process (e.g., uncertainty due to idiosyncratic factors other than the expected price). Assuming naïve expectations (i.e., ∗ = −1), the price dynamics of a reparametrized model can be best represented by: A. Random walk process B. AR(1) process. C. AR(2) process D. Simple Exponential Smoothing 3. The forecast efficiency hypothesis states that forecast errors are mean-zero A. TRUE B. FALSE 4. In an autoregressive model, given by: = + 1−1 + 2−2 + ⋯+ − + , the estimates of 1,2, … , represent the partial autocorrelations of the corresponding lags. A. TRUE B. FALSE 5. A stationary process is also referred to as a white noise process. A. TRUE B. FALSE 6. A random walk is a special case of the autoregressive process. A. TRUE B. FALSE 7. In practice, when comparing multiple forecasting models, the two measures – the mean absolute forecast error (MAFE) and the root mean squared forecast error (RMSFE) – may not always indicate the same preferred model in terms of forecast accuracy. A. TRUE B. FALSE 8. The standard deviation of an h-step-ahead forecast distribution is identical to the standard deviation of the in-sample residuals (assuming no parameter uncertainty) of the stationary AR(1) process. A. TRUE B. FALSE Page 3 of 9 PART B: COMPUTATIONAL QUESTIONS (EACH QUESTION IS WORTH 2 POINTS) Consider the National Activity Index (NAI) – a measure of U.S. economic performance, which we denote by – and the Financial Stress Index (FSI) – a measure of U.S. financial instability, which we denote by . Below are realizations of these variables for the 2018.M11 – 2019.M02 period: t 2018.M11 -0.2 0.2 2018.M12 -0.4 0.1 2019.M01 -0.1 -0.1 2019.M02 0.1 0.2 Suppose, using a total of 100 observations, you estimated an AR(1) model: = + 1−1, ∼ (0,2), where is the NAI in period t, and where the parameter estimates are: � = 0.1 (0.08), ̂ =0.6 (0.23), where the values in parentheses are standard errors of the parameter estimates. Moreover, the estimate of the residual variance: �2 = 0.49. 1. Let denote the kth order autocorrelation, i.e., = (,−). Assuming that the time series is correctly specified by the fitted AR(1) process, fill in the following table with the statistically significant autocorrelations ONLY. 1 2 3 4 5 6 7 8 9 Page 4 of 9 2. Calculate and report the long-run point forecast and its 95% confidence interval. Point forecast ________________ Confidence Interval __________________ Suppose, instead, you hypothesized that financial factors can influence economic performance. Based on this, you estimated the following (autoregressive distributed lag) model that incorporates in the information set: = + 1−1 + 1−1 + , ∼ (0,2), where is the NAI in period t, is the FSI in period t, and where the parameter estimates are: � =0.1 (0.12), ̂ = 0.5 (0.19), � = −0.3 (0.11), where the values in parentheses are standard errors of the parameter estimates. Moreover, the estimate of the residual variance: �2 = 0.42. 3. Based on provided information, does FSI in-sample Granger cause NAI? (briefly explain). Page 5 of 9 4. Briefly describe how would you test whether FSI out-of-sample Granger causes NAI? 5. (note: this question is unrelated to the previous questions) Show that the optimal point forecast under the quadratic loss assumption is the mean. Page 6 of 9 PART C: UNDERSTANDING R (THE ASSIGNMENT IS WORTH 5 POINTS) Below I provide a snippet from an R code. This code has errors. Your job is to flag those errors. For each correctly identified error you receive 1 point. For each incorrectly guessed error (i.e., if you claim there is an error when there is not) you receive -1 point. Your total mark is the sum of these points, bounded by zero and one. That is, if you make only the wrong guesses, you will receive zero points. If you address all the coding errors correctly, you will receive five points. And, for example, if you identify four errors correctly, but also make one incorrect guess, you will receive three (4-1) points. Below is the snippet. You should record your answer on the following page. To give you an example, I report the first error (I label it as “Error 0”, so that you start with “Error 1”). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 # load the 'forecast' package load(forecast) # generate a sequence of 100 standard normal random variables, # denote it by 'x' x <- rnorm(100,0,2) # generate a random walk process, denote it by 'y' y <- cumsum(x) # using 'y' as the dependent variable, estimate an AR(1) model, # denote it by reg.ar lagmat <- embed(y,1) reg.ar <- lm(lagmat[,1]~lagmat[,2]) # using 'y' as the dependent variable, and assuming the rolling # window scheme, where the first rolling window extends from # period 1 to period 50, generate 50 one-step-ahead forecasts from # an AR(1) model; store those forecasts in a vector called 'y.for' y.for <- matrix(NA,50,1) for(i in 1:50){ y.i <- y[1,(i+49)] lagmat.i <- embed(y.i,1) ar.i <- lm(lagmat.i[,1]~lagmat.i[,2]) y.for[i,] <- ar.i$coefficients[1]+ar.i$coefficients[2]*y.i[i+49] } # from these forecasts, obtain the sequence of one-step-ahead forecast errors, denote it by 'e' e <- y-y.for # calculate the root mean squared forecast error measure, # denote it by 'rmsfe' rmsfe <- sqrt(mean(sum(e^2))) Page 7 of 9 Error 0: on line 3, it should be ‘library(forecast)’ or ‘require(forecast)’ instead of ‘load(forecast)’ END OF EXAMINATION
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