ECON 3313: Elementary Economic Forecasting Midterm II Fall 2022
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ECON 3313: Elementary Economic Forecasting
Midterm II
Fall 2022
Problem 1 (40 points)
Suppose we would like to use the historical data on quarterly U.S. airline passengers over the period 1949 Q1 - 1961 Q4 to study the nonstationary components of a time series. We focus only on the trend and seasonality components of the series,
Yt = Tt + St + ϵt (1)
Note that (1) is not an econometric model. We need to further properly model the trend, Tt , and seasonality, St , components.
(1) The airline passengers dynamics feature prominent trend and seasonality. Before you fit any models, (i) discuss whether the trend appears to be linear or nonlinear; (ii) what are the possible explanations for this specific seasonal pattern you observe from this time series plot? [Hint: make sure you answer both (i) and (ii).] (5 points)
(2) In order to fit various trend models, we need to artificially construct the time variable “TIME” . Write all the possible values of TIME for this case. [Hint: recall how you construct TIME in R.] (5 points)
(3) Based on the following model selection criteria, write your preferred econometric model for Tt in (1), i.e, the trend component. Justify your answer. (5 points)
Model |
AIC |
BIC |
Linear Time Trend |
123 2343 |
147 2389 |
Quadratic Time Trend |
135.8971 |
136.7652 |
(4) Answer the following questions regarding model selection criterion.
i. What do most model selection criteria attempt to find? (3 points)
ii. Do you think in-sample MSE is a good criterion for model selection? Briefly explain. (2 points)
iii. If the in-sample MSE is not a good criterion for model selection, how to modify it so that it becomes a good criterion for model selection? Why we need this modification. If the in-sample MSE is a good criterion for model selection, compare the difference between the in-sample MSE, AIC, and BIC. (3 points)
iv. How does in-sample MSE relate to R2 , AIC, and BIC? Briefly explain. (5 points)
v. Discuss the difference and the relationship (similarity) between R2 and adjusted R2 . Which one is better for model selection? Why? (5 points)
(5) In this case, how would you build an econometric model for the seasonal component St in (1)? Write the full econometric model for yt using your preferred model of St and your preferred model of Tt in part 1.3. [Hint: you need to write two econometric models, one for St and one for yt .] Please also indicate specifically how many seasonal components are needed, whether the intercept should be included in the model, and briefly explain the reason. (7 points)
Problem 2 (60 points)
Consider the following time series process (i.e., linear time trend process),
yt = β0 + β1 t + ϵt , (2)
where ϵt N(0,σ2 ) and σ 2 < ∞ .
(1) Is ϵt a white noise? Please verify your answer using the definition of white noise to receive the full credit. (6 points)
(2) Now consider a new process xt constructed from
xt = yt − 0.75yt − 1 − 0.25yt −2
Please derive the mean and variance of xt (i.e., E[xt] and Var(xt )). Do they depend on t? (14 points)
(3) Assume we only have the information until time t − 2, i.e., the information set is Ωt −2 = {yt −2 ,yt −3 , ··· }.
Please derive the conditional mean and conditional variance of xt based on Ωt −2 (i.e., E[xt |Ωt −2 ) and Var(xt |Ωt −2 )). (10 points)
(4) The autocovariance of the new process xt for any displacement τ is defined as γ(t,τ) = cov(xt ,xt −τ ) = E[xt xt −τ ] − E[xt]E[xt −τ ]
Note: this covariance formula is only true for (3).
Derive the autocovariance function γ(t,τ) for xt , where τ ≥ 0. Does it depend on t? Hint: discuss four cases: τ = 0, τ = 1, τ = 2, and τ > 2. (20 points)
(4) (continued)
(5) Please rewrite xt by using the lag operator. (10 points)
2022-12-12