4. Time Series Flashcards
A stochastic process is said to be weakly stationary if…
- Means equal
- Covariances equal
As a consequence, also stationary in the variance
How can we identify stationarity? Without charts
We study successive samples of moderate size.
Weak: sample means and Std devs to be approximately the same
Strong: distributions to be roughly the same
How can we use control charts to look at stationarity?
Using UCL and LCL (using mean and sy from 1,…,n)
X-bar chart: helps examine stationarity of the mean
R chart: examine stationarity of the variance by plotting successive ranges of the successive samples.
Weak stationarity: points within bounds
No stationarity: points outside bounds
How can we make a process stationary if it is not?
- Differencing - stationary mean
- Transformations Ln(y) - stationary variance
If both are done, we have a process stationary in the mean and variance
What is a white noise process? Does the distribution change with time?
It is a sequence of iid variables. The distribution does not change with time
What is the l-step ahead forecast of Yn? What is the standard error of this? WN
Y-hat = y-bar
Standard error = standard deviation of y times the square root of (1+1/n)
What is the prediction interval for l-step ahead forecast of Y if it belongs to a white noise process? Do these prediction intervals increase with time?
Formulas. No, they stay constant over time because the Y’s are not increasing with time
How would you model a random walk ? Yt = ______
Formula
What is a random walk process defined as?
Partial sums of a white noise process
What is the expected value and variance of a RW process?
Formula
Is a random walk stationary? When would the process be stationary in the mean?
No, because it’s mean and variance increase with time. The process would be stationary in the mean if the mean of the WN process had a mean of 0. (Mu=0)
What is the l-step ahead forecast of a random walk process? What is its standard error?
Formula
What is the 95% prediction interval for the l-step ahead forecast of a random walk?
Formula
Use 2 as t quantile always
How could we find w-bar if we are not given all values of a random walk? (Only given first and last 5 values of the time series)
Formula
How could you identify a random walk process? 3
- Examine the series
Mean should be linear and the variance should increase with time - Examine the differenced series
White noise process - Compare standard deviations of the random walk and differenced series
RWstddev»_space; WNstddev
What are the 5 statistics for comparing time series models (WN and RW)?
Formula
What is the formula for a lag-k autocorrelation?
Formula
What is meant if a time series has a meandering process?
This means that the lag-1 autocorrelation has a value greater than 0.
If we would like to test that the lag-k autocorrelation is different from 0, what test statistic would we use? From what distribution?
Formula, normal distribution
What are the 3 assumptions for an AR(1) model?
- Expected value of error = 0
- Variance of error = sigma
- Corr(E_t+k, Y_t) = 0
If we have a stationary AR(1)model, what are the estimates for the expected value, variance and lag-k autocorrelation?
Formula
How could we identify auto regression?3
- Check if the series is stationary
- Plot adjacent values
They should be linearly related - Check it’s autocorrelations
They should be a decreasing geometric series (their absolute values)
What are the estimates of beta1 and beta0 in an AR(1) model? What method is used?
Formula and CLS
How could we approximate the estimates for beta1 and beta0 in an AR(1) model?
Formula
What is the estimate for the models variance in an AR(1) model?
MSE Formula df= n-3
What is the l-step ahead forecast of an AR(1) model? What is its standard error? What distribution and df is used in the prediction interval?
Formula
T distribution with df =n-3
What is the moving average estimate at time t? Recursively?
Formula
What is the formula for the doubly smoothed moving average?
Formula
How do we forecast with moving average and doubly smoothed moving averages?
Formula
What is the formula for the exponential smoothed estimate? Recursively?
Formula
What is the formula for the doubly smoothed series with exponential smoothing
Formula
How would we forecast with exponential smoothing? Both single and doubly smoothed
Formula
Is smoothing appropriate for time series data that shows a linear trend?
No, only appropriate for time series data without a linear trend
What is smoothing related to? What other regression technique?
Weighted least squares
When there is a linear trend in time, what type of smoothing can be used for predictions?
Doubly smoothed
What is a unit root test used for?
Evaluates the fit of a random walk model
What are two examples of unit root tests?
Dickey-fuller test and the augmented dickey-fuller test
What is an ARCH model used to model?
It’s used to model the conditional variance of a time series, not the time series itself.
Is an AR(1) model a meandering process?
Only if the slope coefficient is positive
Is a stationary AR(1) process a generalization of a white noise and a random walk?
No, a GENERAL AR(1) model is a generalization of a WN and RW. Not stationary because the beta1 cannot equal 1 (cannot be a random walk then)
