Chapter 6 question from Cambridge

A white noise will have a. A zero mean b. A constant variance c. Autocovariances that are constant d. Autocovariances that are zero except at lag zero

A, B, C

Which of the following conditions are necessary for a series to be classifiable as a weakly stationary process? a. It must have a constant mean b. It must have a constant variance c. It must has constant autocovariances for given lags d. It must have a constant probability distribution

A, B, C

Which of the following statements are true concerning the acf and pacf? a. The acf and pacf are often hard to interpret in practice b. The acf and pacf can be difficult to calculate for some data sets c. Information criteria represent an alternative approach to model order determination d. If applied correctly, the acf and pacf will always deliver unique model selections

A, B, C

Which of the following statements are true concerning the autocorrelation function (acf) and partial autocorrelation function (pacf)? a. The acf and pacf will always be identical at lag one whatever the model b. The pacf for an MA(q) model will in general be non-zero beyond lag q c. The pacf for an AR(p) model will be zero beyond lag p d. The acf and pacf will be the same at lag two for an MA(1) model

A, B, C. The pacf measures the correlation between y_t and y_(t-k) after controlling for (removing) the effects of the intermediate lags on the current value. For example, the pacf at lag 3 measures corr(y_t, y_(t-3)) after removing the effects of y_(t-1) and y_(t-2) on y_t. Therefore, since at lag 1 there are no intermediate lags to remove, the acf and pacf will always be identical at lag 1 whatever the model, so (i) is correct. For an MA(q) model, the acf will be zero at all lags beyond q, but the MA(q) can be written as an AR(infinity). Therefore, the pacf will never be zero, but will decline geometrically, and thus (ii) is correct. For an AR(p), however, once the effects of y_(t-1), y_(t-2), ..., y(t-p) are removed, the correlation between y_t and y_(t-p-j) will be zero for all positive integer values of j. So, whilst the acf for an AR(p) will decline geometrically, the pacf will be zero after p lags and thus (iii) is true. Finally, although the acf will be zero at lag 2 for an MA(1), the pacf will not so (iv) is false.

Assuming that the coefficients are approximately normally distributed, which of the coefficients are statistically significant at the 5% level? A. 1 only; B. 1 and 2 only c. 1, 2, and 3 only d. It is not possible to determine the statistical significance since no standard errors have been given

B. 1 and 2 only

The (unconditional) variance of the AR(1) process for y given yt = 0.2 + 0.4 yt-1 + ut a. 1.19 b. 2.5 c. 1 d.33

The (unconditional) variance of an AR(1) process is given by the variance of the disturbances divided by (1 minus the square of the autoregressive coefficient), which in this case is 1 / (1 - 0.4^2) = 1.19

The value of the autocorrelation function at lag 3 for the AR(1) model given in question 12 will be yt = 0.2 + 0.4 yt-1 + ut a. .4 b. .064 c. 0 d. .076

The value of the autocorrelation function at lag k for any AR(1) process with autoregressive coefficient a1 is simply given by a1^k, which in this case is 0.4^3 = 0.064. b. .064

The value of the autocovariance function at lag 3 for the AR(1) model given in question 12 will be yt = 0.2 + 0.4 yt-1 + ut a. .4 b. .064 c. 0 d. .076

The value of the autocovariance function at lag k for any AR(1) process with autoregressive coefficient a1 is given by a1^k multiplied by sigma^2 divided by (1 minus a1^2) , which in this case is 0.4^3 x 1 / (1- 0.4^2) = 0.076. d. .076

An ARMA(p,q) (p, q are integers bigger than zero) model will have a. An acf and pacf that both decline geometrically b. An acf that declines geometrically and a pacf that is zero after p lags c. An acf that declines geometrically and a pacf that is zero after q lags d. An acf that is zero after p lags and a pacf that is zero after q lags

a. An acf and pacf that both decline geometrically

Which of the following statements are true concerning information criteria? a. Adjusted R-squared is an information criterion b. If the residual sum of squares falls when an additional term is added, the value of the information criterion will fall c. Akaike's information criterion always leads to model orders that are at least as large as those of Schwarz's information criterion d. Akaike's information criterion is consistent

a. Adjusted R-squared is an information criterion c. Akaike's information criterion always leads to model orders that are at least as large as those of Schwarz's information criterion

If a series, y, follows a random walk, what is the optimal one-step ahead forecast of y? a. The current value of y b. zero c. one d. The average value of y over the in-same period

a. The current value of y

The characteristic roots of the MA process yt = -3ut-1 + ut-2 + ut a. 1 and 2 b. 1 and .5 c. 2 and -.5 d. 1 and -3

b. 1 and .5 The roots of the characteristic equation are found for an MA process by first using the lag operator notation and gathering all of the terms in u together as y_t = -3L u_t + L^2 u_t + u_t. Then the characteristic equation will be z^2 - 3z + 1 = 0, which factorises to (1 - z)(1 - 2z) = 0, giving roots of 1 and 0.5 (so b is correct). Out of interest, this MA process is non-invertible since invertibility would require both roots to lie outside the unit circle while in this case there is one unit root and one explosive root. This MA process would therefore "blow up" under the AR(infinity) representation with the coefficients on the terms getting bigger and bigger on the lags further and further back into the pas

The pacf is necessary for distinguishing between a. An AR and an MA model b. An AR and an ARMA model c. An MA and an ARMA model d. Different models from within the ARMA family

b. An AR and an ARMA model

Which of the following statements are true concerning the class of ARIMA(p,d,q) models? a. The "I" stands for independent b. An ARIMA(p,1,q) model estimated on a series of logs of prices is equivalent to an ARIMA(p,0,q) model estimated on a set of continuously compounded returns c. It is plausible for financial time series that the optimal value of d could be 2 or 3 d. The estimation of ARIMA models is incompatible with the notion of cointegration

b. An ARIMA(p,1,q) model estimated on a series of logs of prices is equivalent to an ARIMA(p,0,q) model estimated on a set of continuously compounded returns d. The estimation of ARIMA models is incompatible with the notion of cointegration

Which of the following statements is INCORRECT concerning a comparison of the Box-Pierce Q and the Ljung-Box Q* statistics for linear dependence in time series? a. Asymptotically, the values of the two test statistics will be equal b. The Q test has better small-sample properties than the Q*. c. The Q test is sometimes over-sized for small samples d. As the sample size tends towards infinity, both tests will show a tendency to always reject the null hypothesis of zero autocorrelation coefficients

b. The Q test has better small-sample properties than the Q*.

Consider the following MA(3) process yt = μ + εt + θ1εt-1 + θ2εt-2 + θ3εt-3 , where εt is a zero mean white noise process with variance s2. Which of the following statements are true? a. The process yt has zero mean b. The autocorrelation function will have a zero value at lag 5 c. The process yt has variance s2 d. The autocorrelation function will have a value of one at lag 0

b. The autocorrelation function will have a zero value at lag 5 d. The autocorrelation function will have a value of one at lag 0

Which of the following statements are true concerning the Box-Jenkins approach to diagnostic testing for ARMA models? a. The tests will show whether the identified model is either too large or too small b. The tests involve checking the model residuals for autocorrelation, heteroscedasticity, and non-normality c. If the model suggested at the identification stage is appropriate, the acf and pacf for the residuals should show no additional structure d. If the model suggested at the identification stage is appropriate, the coefficients on the additional variables under the overfitting approach will be statistically insignificant

b. The tests involve checking the model residuals for autocorrelation, heteroscedasticity, and non-normality d. If the model suggested at the identification stage is appropriate, the coefficients on the additional variables under the overfitting approach will be statistically insignificant

Consider the following AR(2) process: yt = 1.5 yt-1 - 0.5 yt-2 + ut a. Stationary process b. Unit root process c. Explosive process d. Stationary and unit root process

b. Unit root proce

Consider again the autocorrelation coefficients described in question 5. The value of the Box-Pierce Q-statistic is a. .12 b. 37.50 c. 18.12 d. 18.09

c. 18.12 Q = 250* (0.2^2 + -0.15^2 + 0.1^2) = 18.12.

For an autoregressive process to be considered stationary a. The roots of the characteristic equation must all lie inside the unit circle b. The roots of the characteristic equation must all lie on the unit circle c. The roots of the characteristic equation must all lie outside the unit circle d. The roots of the characteristic equation must all be less than one in absolute value

c. The roots of the characteristic equation must all lie outside the unit circle

Consider the following AR(1) model with the disturbances having zero mean and unit variance yt = 0.2 + 0.4 yt-1 + ut The (unconditional) mean of y will be given by a. .2 b. .4 c. .5 d. .33

d. .33 For an AR(1) process, the (unconditional) mean of y will be given by the intercept divided by (1 minus the autoregressive coefficient), which in this case is 0.2 / (1-0.4) = 0.33

Which of the following statements is true concerning forecasting in econometrics? a. Forecasts can only be made for time-series data b. Mis-specified models are certain to produce inaccurate forecasts c. Structural forecasts are simpler to produce than those from time series models d. In-sample forecasting ability is a poor test of model adequacy

d. In-sample forecasting ability is a poor test of model adequacy

Consider a series that follows an MA(1) with zero mean and a moving average coefficient of 0.4. What is the value of the autocovariance at lag 1? a. .4 b. 1 c. .34 d. It is not possible to determine the value of the autocovariances without knowing the disturbance variance

d. It is not possible to determine the value of the autocovariances without knowing the disturbance variance

If a series, y, follows a random walk with drift b, what is the optimal one-step ahead forecast of the change in y? a. The current value of y b. zero c. one d. The average value of the change in y over the in-sample period

d. The average value of the change in y over the in-sample period