Indian Statistical Service Examination (with Indian Economic Service) · India · Union Public Service Commission
Original practice questions written to the published syllabus · not past papersModelling observations ordered in time, from classical decomposition to Box-Jenkins and beyond: stationarity, ARIMA and seasonal ARIMA, volatility models for prices, vector autoregressions, state-space models and the Kalman filter, and forecast evaluation. Agribusiness econometrics introduces ARIMA and unit roots; this subject is the full treatment.
In the exams: AICE-JRF Ag Stat Unit 2 'components of time series' (present); UPSC ISS Statistics III/IV (heavy, reported); RBI Grade B DSIM (heavy, reported); PhD course in most programmes
How our questions for Indian Statistical Service Examination spread across the units of Time Series Analysis. Difficulty: 27 easy, 50 medium, 22 hard.
| AR, MA and ARMA processes | 11 | |
| Unit-root tests and structural breaks | 10 | |
| Vector autoregression and cointegration | 10 | |
| Forecast accuracy: MAPE, RMSE and the Diebold-Mariano test | 10 | |
| Stationarity, autocorrelation and partial autocorrelation functions | 9 | |
| Box-Jenkins ARIMA: identification, estimation and diagnostic checking | 9 | |
| Seasonal ARIMA and ARIMAX with weather covariates | 9 | |
| ARCH and GARCH models for price volatility | 9 | |
| State-space models and the Kalman filter | 8 | |
| Spectral analysis (introduction) | 7 | |
| Smoothing: moving averages and exponential smoothing (Holt-Winters) | 6 | |
| Components of time series and classical decomposition | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
The sample ACF of a monthly price series starts near 0.95 and declines very slowly and almost linearly over 30 lags. What should the analyst do first?
A. Difference the series
A slow, nearly linear decay is the signature of non-stationarity (a unit root); the series should be differenced (and tested formally) before identifying ARMA orders. An MA(30) is grossly over-parameterised, and the ACF clearly rules out white noise.
Further reading: Box, Jenkins, Reinsel and Ljung, Time Series Analysis
A stationary series has rho(1) = 0.6 and rho(2) = 0.5. What is the partial autocorrelation at lag 2?
D. 0.219
phi(2,2) = (rho2 - rho1^2)/(1 - rho1^2) = (0.5 - 0.36)/0.64 = 0.219. Forgetting the denominator gives 0.140; 0.500 is the ordinary autocorrelation; 0.360 is rho1 squared, the lag-2 ACF an AR(1) would imply.
Further reading: Box, Jenkins, Reinsel and Ljung, Time Series Analysis
Why is a random walk, Y(t) = Y(t-1) + e(t) with e(t) white noise, not stationary even though its expected change is zero?
A. Its variance grows with time
Starting from Y(0), Var(Y(t)) = t sigma^2, which grows without limit, so the variance is not constant. The mean stays at Y(0) with no drift; the shocks are white noise; and the first difference is white noise, which is stationary.
Further reading: Box, Jenkins, Reinsel and Ljung, Time Series Analysis
The process Y(t) = 4 + 0.6 Y(t-1) + e(t) is stationary. What is its mean?
B. 10
mu = c/(1 - phi) = 4/0.4 = 10. 4 is the constant, not the mean; 6.67 = 4/0.6; 2.4 = 4 x 0.6.
Further reading: Box, Jenkins, Reinsel and Ljung, Time Series Analysis; Brockwell and Davis
What is the order of the stages in the Box-Jenkins approach to model building?
A. Identification, estimation, diagnostic checking, then forecasting
Box and Jenkins proposed an iterative cycle: identify a tentative model, estimate it, check the residuals, and return to identification if the checks fail; forecasting follows once the model is adequate.
Further reading: Box, Jenkins, Reinsel and Ljung, Time Series Analysis
A regression of one district's wheat price on an unrelated district's cotton area, both trending random walks, gives R-squared = 0.92 and a Durbin-Watson statistic of 0.25. What is the most likely explanation?
D. A spurious regression between non-stationary series
Granger and Newbold showed that regressing independent random walks gives high R-squared and very low Durbin-Watson; R-squared above DW is a warning sign. The fit says nothing about a real relation unless the series are cointegrated.
Further reading: Enders, Applied Econometric Time Series; Hamilton, Time Series Analysis
A GARCH(1,1) model for weekly returns has omega = 0.2, alpha = 0.10 and beta = 0.85. What is the unconditional variance of returns?
C. 4.0
Unconditional variance = omega/(1 - alpha - beta) = 0.2/0.05 = 4.0. Dividing by 1 - beta only gives 1.33; dividing by alpha + beta gives 0.211.
Further reading: Enders, Applied Econometric Time Series
Engle's LM test regresses squared residuals from a price-return model on 4 of their own lags, with n = 200 and R-squared = 0.06. The 5 per cent critical value of chi-square with 4 df is 9.49. What is the conclusion?
A. LM = 12; ARCH effects are present
LM = n R-squared = 200 x 0.06 = 12 > 9.49, so the null of no ARCH is rejected. Dividing by the number of lags or using R-squared x 100 gives 3 or 6, both below the critical value, and wrongly accepts the null.
Further reading: Enders, Applied Econometric Time Series
Two I(1) price series are said to be cointegrated when:
B. Some linear combination of them is stationary
Cointegration means a linear combination is I(0), so the series share a long-run equilibrium. Being I(1) individually is the precondition, not the definition; high correlation and Granger causality are neither necessary nor sufficient.
Further reading: Enders, Applied Econometric Time Series; Hamilton, Time Series Analysis
What does the spectrum of white noise look like?
B. Flat across all frequencies
White noise has equal variance contribution at every frequency, a flat spectrum. A peak at zero frequency is typical of strongly persistent series.
Further reading: Chatfield, The Analysis of Time Series; Brockwell and Davis
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