Section 1: MCQ/MAQ/TFQ
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, 𝑞𝑞𝑡𝑡 = 𝛼𝛼 − 𝛽𝛽𝑝𝑝𝑡𝑡, and the
supply is given by 𝑞𝑞𝑡𝑡 = 𝛾𝛾 + 𝛿𝛿𝑝𝑝𝑡𝑡∗ + 𝜀𝜀𝑡𝑡, where 𝑝𝑝𝑡𝑡∗ is the expected price, and where 𝜀𝜀𝑡𝑡 is an iid
white noise process (e.g., uncertainty due to idiosyncratic factors other than the expected
price). Assuming naïve expectations (i.e., 𝑝𝑝𝑡𝑡∗ = 𝑝𝑝𝑡𝑡−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: 𝑦𝑦𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽1𝑦𝑦𝑡𝑡−1 + 𝛽𝛽2𝑦𝑦𝑡𝑡−2 + ⋯+ 𝛽𝛽𝑝𝑝𝑦𝑦𝑡𝑡−𝑝𝑝 + 𝜀𝜀𝑡𝑡, the
estimates of 𝛽𝛽1,𝛽𝛽2, … ,𝛽𝛽𝑝𝑝 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
David Ubilava
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Section 2: Numerical
Consider the National Activity Index (NAI) – a measure of U.S. economic performance, which we
denote by 𝑦𝑦 – and the Financial Stress Index (FSI) – a measure of U.S. financial instability, which we
denote by 𝑧𝑧. Below are realizations of these variables for the 2018.M11 – 2019.M02 period:
t 𝑦𝑦𝑡𝑡 𝑧𝑧𝑡𝑡
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:
𝑦𝑦𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽1𝑦𝑦𝑡𝑡−1𝜀𝜀𝑡𝑡, 𝜀𝜀𝑡𝑡 ∼ 𝑖𝑖𝑖𝑖𝑖𝑖(0,𝜎𝜎2),
where 𝑦𝑦𝑡𝑡 is the NAI in period t, and where the parameter estimates are: 𝛼𝛼� = 0.1 (0.08), �̂�𝛽 =
0.6 (0.23), where the values in parentheses are standard errors of the parameter estimates.
Moreover, the estimate of the residual variance: 𝜎𝜎�2 = 0.49.
1. Let 𝝆𝝆𝒌𝒌 denote the kth order autocorrelation, i.e., 𝝆𝝆𝒌𝒌 = 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪(𝒚𝒚𝒕𝒕,𝒚𝒚𝒕𝒕−𝒌𝒌). 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.
𝑘𝑘 1 2 3 4 5 6 7 8 9
𝜌𝜌
David Ubilava
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1.2 Calculate and report the long-run point forecast and its 95% 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
𝑧𝑧 in the information set:
𝑦𝑦𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽1𝑦𝑦𝑡𝑡−1 + 𝛾𝛾1𝑧𝑧𝑡𝑡−1 + 𝜀𝜀𝑡𝑡, 𝜀𝜀𝑡𝑡 ∼ 𝑖𝑖𝑖𝑖𝑖𝑖(0,𝜎𝜎2),
where 𝑦𝑦𝑡𝑡 is the NAI in period t, 𝑧𝑧𝑡𝑡 is the FSI in period t, and where the parameter estimates are: 𝛼𝛼� =
0.1 (0.12), �̂�𝛽 = 0.5 (0.19), 𝛾𝛾� = −0.3 (0.11), where the values in parentheses are standard errors of
the parameter estimates. Moreover, the estimate of the residual variance: 𝜎𝜎�2 = 0.42.
1.3 Based on provided information, does FSI in-sample Granger cause NAI? (briefly explain).
Suppose, instead, you have quarterly data from a little bit earlier period. You estimated a basic AR(1) model
which yielded the parameter estimates of -0.04 and 0.68 for intercept and the lagged dependent variable
respectively, and the residual standard deviation estimate of 0.65.
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Consider the following (additional) information:
t yt
2016.Q3 -0.22
2016.Q4 -0.22
2017.Q1 -0.67
2017.Q2 0.05
Suppose observations up to and including 2016.Q4 have been used in parameter estimation:
2.3 Calculate and report the one-step-ahead point forecasts and the associated the 95% interval
forecasts of the NAI in the subsequent two months.1
Suppose, now, that you fitted a random walk model:
𝑦𝑦𝑡𝑡 = 𝑦𝑦𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
where, the estimate of the residual standard error is: 𝜎𝜎� = 0.70.
2.4 (2pt): Calculate and report the one-step-ahead point forecasts and the associated 95% interval
forecasts of the NAI in the subsequent two months.
Suppose, now, that you fitted a seasonal component model:
𝑦𝑦𝑡𝑡 = �𝛿𝛿𝑖𝑖𝑑𝑑𝑡𝑡𝑖𝑖
𝑖𝑖
+ 𝜀𝜀𝑡𝑡
where 𝑑𝑑𝑡𝑡𝑖𝑖, (𝑖𝑖 = 1, … ,4), are quarterly binary variables, representing quarters 1 – 4, respectively. The
parameter estimates are: 𝛿𝛿1 = −0.14, 𝛿𝛿2 = −0.15, 𝛿𝛿3 = −0.20, 𝛿𝛿4 = −0.06; and 𝜎𝜎� = 0.90.
2.5 (2pt): Calculate and report the one-step-ahead point forecasts and the associated the 95% interval
forecasts of the NAI in the subsequent two months.
2.6 (2pt): Using the point forecasts from questions 2.3, 2.4 and 2.5, obtain the root mean square
forecast error (RMSFE) measures for each of the three considered models and identify the preferred one.
________
quarters.________
quarters.
________ quarters.
Numerical Questions – (work on your own)
1. 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 yt zt
2018.M11 -0.4 0.1
2018.M12 -0.2 -0.1
2019.M01 0.1 0.2
2019.M02 -0.2 0.1
Suppose, using a total of 100 observations, you estimated an AR(2) model:
yt = α + β1yt-1 + β2yt-2 + εt, where εt∼iid(0,σ2)
which resulted in the following summary of the regression outputs:
estimation window
parameters 1991.M01-2018.M12 1991.M02-2019.M01
α1 0.06 0.05
β1 0.35 0.33
β2 0.46 0.44
σ 0.55 0.54
A. (1pt) Based on provided information, from first estimation window, calculate and
report the one-step-ahead and the two-step-ahead point forecasts of y.
B. (1pt) Based on provided information, from first estimation window, calculate and
report the one-step-ahead and the two-step-ahead 95% interval forecasts of y.
C. (1pt) Based on provided information, can you tell (and if so, do tell) whether you
have used recursive, rolling, or fixed window approach for generating the out-of-
sample forecasts?
It could be that the autoregressive dynamics of y is regime-dependent. That is, the
dynamics of y may change depending on lagged value of z. With that in mind,
suppose you estimated the following second order threshold autoregressive model:
yt = (α + β1yt-1 + β2yt-2)I(zt-1 ≤ 0)+ (θ + δ1yt-1 + δ2yt-2)I(zt-1 > 0)+ εt,
where εt∼iid(0,σ2), and where I(·) is a Heaviside indicator function that takes on one
if the condition within the parentheses is satisfied, and zero otherwise. Suppose this
resulted in the following summary of the regression outputs:
estimation window
parameters 1991.M1-2018.M12 1991.M2-2019.M01
α 0.04 0.03
β1 0.18 0.16
β2 0.36 0.40
θ -0.02 -0.01
δ1 0.24 0.26
δ2 0.16 0.12
σ 0.52 0.53
D. (2pt) Based on provided information, for both estimation windows, calculate and
report the one-step-ahead point forecasts of y.
2. Consider annual time series of wheat production (metric tonnes per hectare),
denoted by Yt. Suppose you estimated the following nonlinear trend model:
Yt = α + β1t + β2(t-τ)I(t-τ) + ε t, where t = 1,…,T; and ε ∼ iid(0,σ2), where ,σ2=0.64.
I(·) is a Heaviside indicator function that takes on one if the condition within the
parentheses is satisfied, and zero otherwise. Suppose T=99, and τ=50, and the
parameter estimates for α, β1, and β2 are 1.2, 0.08 and -0.05, respectively.
Calculate and report the point forecast and interval forecast of YT+1.
ECMT3130_Midterm_Practice.pdf
ECMT3130_Midterm_Practice_2.pdf
ECMT3130_Final_Practice.pdf