Time Series

Question 1

In order for an AR(1) model to exhibit STATIONARITY, what are the bounds of the B1 coefficient?

Answer 1

B1 has to greater than -1 and less than 1, i.e. the absolute value of the B1 coefficient has to be less than 1.

Question 2

Describe what the AR(1) model becomes when B1=0 or B1=1.

Answer 2

If B1=0, then Y follows a white noise process. If B1=1, then Y follows a random walk process.

Question 3

Given a series of observations, how do we know if an autoregressive model is suitable for modeling the data?

Answer 3

1. Use control charts to detect if the series exhibits stationarity (needs to be).
2. Plot adjacent values to see if adjacent values are linearly related.
3. Check the absolute value of its lag k autocorrelation to see if it forms a decreasing geometric sequence as k increases.

Question 4

What weight must we assign in exponential smoothing to result in no smoothing?

Answer 4

w=0

Question 5

How do we choose the optimal value for w?

Answer 5

We calculate SS(w), which is the sum of squared one-step prediction errors, for all different values of w, and choose the value that results in the lowest SS(w).

Question 6

What is another name for the estimates calculated from exponential smoothing?

Answer 6

Discounted least squares estimates

Question 7

What are three indications that a nonstationary times series can be represented as a random walk?

Answer 7

1. A control chart of the series detects a linear trend in time and increasing variability.
2. The differenced series follows a white noise model.
3. The standard deviation of the original series is greater than the standard deviation of the differenced series.

Question 8

T/F: It is risky to choose a small value for k
because we may lose sight of the real trends due to oversmoothing.

Answer 8

False. A small k doesn’t cause oversmoothing. It is risky to choose a large value for k
because we may oversmooth the data and lose sight of the real trends.

Question 9

T/F: When smoothing with moving averages is used for forecasting, the model is called a globally constant mean model.

Answer 9

False. Smoothing with moving averages results in a locally constant mean model. The model is only called a globally constant mean model when equal weight is given to all observations.

Question 10

What is the purpose of a unit root test?

Answer 10

A unit root test is used to evaluate the fit of a random walk model. A random walk model is a good fit if the time series possesses a unit root. The Dickey-Fuller test and augmented Dickey-Fuller test are two examples of unit root tests.

Question 11

What is the lag k autocorrelation of a white noise process?

Answer 11

0

Question 12

What is the ARCH model used for?

Answer 12

An ARCH model is used to model the conditional variance of a time series, not the time series itself

Question 13

T/F: A control chart has superimposed lines called reference limits.

Answer 13

False: Control charts have superimposed lines called control limits. Two well-known control limits are the upper control limit and the lower control limit.

Question 14

T/F: An R chart examines the stability of the mean of a time series.

Answer 14

False. An R chart helps examine the stability of the variability of a time series. An x-chart examines the stability of the mean of a time series.

Question 15

T/F: An AR(1) model is a meandering process.

Answer 15

False. Not all AR(1) model is a meandering process. An AR(1) model with −1<β1<0 is not a meandering process. An AR(1) model with 0<β1<1 is a meandering process. (i.e. meandering if B1 is stationary AND positive)

Question 16

T/F: A stationary AR(1) model is a generalization of both a white noise process and a random walk model.

Answer 16

False. A general AR(1) model is a generalization of a white noise process and a random walk model. However, note that the slope coefficient of a stationary AR(1) model is restricted to be between -1 and 1 exclusively, i.e. −1<β1<1. This means that a stationary AR(1) model is not a generalization of a random walk model since β1=1
for a random walk model.

Question 17

T/F: The lag k autocorrelation of a stationary AR(1) model is always non-negative.

Answer 17

False. This is only true if the slope coefficient is non-negative, i.e. β1≥0. If it is negative, then the lag k autocorrelation will be negative when k is odd. Recall that a stationary AR(1) model has a lag k autocorrelation of ρk=βk1.
.

Question 18

What are the three assumptions upheld by the AR(1) model?

Answer 18

1. The expected mean of the error term is 0.
2. The variance of the error term is sigma^2.
3. The covariance of the future error term and current Y is 0, i.e. future error is independent of current value.