PS5_sol(1).pdf - Problem Set 5 Spectra and Introduction to...

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Problem Set 5: Spectra and Introduction to Nonstationarity ECON 21200, Spring 2015 SOLUTIONS Problem 1 (Spurious Regressions) Finite samples can often give misleading results, especially if the time series are highly persistent. In this problem we will explore how two independent series can appear to be highly correlated. Start by simulating Y t = 0 . 99 Y t - 1 + t and X t = 0 . 99 X t - 1 + ν t each for 100 time periods. Repeat the simulation 10,000 times. Let t , ν t be iid N (0 , 1) , and also independent of each other. Initiate Y 0 = X 0 = 100 (the unconditional mean of both processes). a) What is the actual value of γ in the regression Y t = γX t + δ t ? Solution: Since Y t and X t are completely independent, γ = 0. b) Regress your simulated values of Y t on your simulated values of X t (no constant) and store the value of ˆ γ for each of your 10,000 simulations. Plot a histogram of the values. Solution: See Figure 1. c) What is the mean of the distribution of ˆ γ ? Is this what you expected based on part (a)? Solution: From my simulations, the mean of the distribution is -0.0025, which is basically what we expected. d) What percentage of the values are significant at the 5% level? Is this what you expected? If you need to calculate significance, remember that you can use the OLS matrix formula Var ˆ β = σ 2 ( X 0 X ) - 1 . Solution: From CLT we know ˆ γ is normally distributed. So if you standardize the distribution to get z -scores, you can count how many exceed the 5% cutoff of 1.96. In my sample, it was 9420 out of 10000, which means 1
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-5 -4 -3 -2 -1 0 1 2 3 4 0 100 200 300 400 500 600 ˆ γ count Distribution of estimates of γ Figure 1: Distribution of ˆ γ , question 1 part (b) the vast majority of the regressions imply that Y t and X t have some statistically significant relationship. Thus, we are led to conclude incorrectly that Y t and X t are not independent. e) Reflect briefly on the lessons from this problem. E.g. what does it imply for actual research, and what are some appropriate ways of handling data to avoid false conclusions? The problem clearly illustrates the importance of drawing reasonable qualitative conclusions about your data and not trusting statistical significance. For example,
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