QE 7/8 - time series Flashcards
Difference between a predicted value and a forecast?
- Predicted value – refers to value of Y predicted (using regression) for observation WITHIN SAMPLE used to estimate regression
- Forecast – refers to value of Y forecasted for observation OUT OF SAMPLE used to estimate regression
Difference between a forecast error and OLS residual?
- OLS residual = within sample (difference between predicted value and actual value)
- Forecast error = same concept but out of sample
What does RMSFE measure?
- Measures spread of forecast error distribution
2. Measures magnitude of typical forecasting ‘mistake’
Sources of error in the RMSFE
(1) future values of u unknown
2) error in estimating coefficients (B0 & B1
When is RMSFE not an appropriate measure of the magnitude of a typical forecasting mistake? Example?
- If forecasting mistakes asymmetric
- E.g. when forecasting time I’ll arrive at train station, under-forecast (being late) much worse than over-forecast (being early)
How to test the hypothesis that, say, regressors Yt-2, Yt-3,…,Yt-p don’t further help forecast (beyond Yt-1)?
- F-test that coefficients all jointly zero
- Information criterion (BIC or AIC)
(i) E.g. Bayes information criterion (BIC) determines how large the increase in R-squared must be to justify including the additional lag
What is the Granger causality test?
- Test of joint hypothesis that none of X’s a useful predictor, above and beyond lagged values of Y
- i.e. F-statistic testing hypothesis that coefficients on all values of 1 of variables are zero (implying regressors have no predictive content for Yt beyond that contained in other regressors)
- N.B. NOT a test of causality (causality here just refers to predictive content)
What is the trade-off of using additional lagged values as predictors?
- Too few lags decreases forecast accuracy because valuable information is lost
- Too many lags increases estimation uncertainty
Generally, an AR(…..) in 1st difference = AR(…..) in level
Generally, an AR(p) in 1st difference = AR(p+1) in level
- What does it mean for Yt to have very strong autocorrelation?
- What is the consequence of this?
- What happens in the extreme case when autocorrelation = 1?
- Possible solution?
- Very persistent process
- OLS estimator of the AR coefficient is biased towards zero
- In the extreme case, Yt no longer stationary
- Take 1st differences
- What does Granger causality mean?
2. Granger non-causality?
- Granger causality - at least 1 of the coefficients of the lags of X is not zero
- Granger non-causality - all the coefficients of the lags on X are zero
What is the only way to remove a stochastic trend? Exception?
Only way to remove a stochastic trend is by differencing, unless there’s co-integration
Problems caused by stochastic trends/unit root?
- Autoregressive coefficients biased downwards towards zero
- Distribution of OLS estimator and t-statistic not normal, even in large samples
- Spurious regression
Explain how ‘stochastic trend’ and ‘unit root’ can be used interchangeably?
- If Yt has a unit root, then Yt contains a stochastic trend (and so is non-stationary)
- If Yt is stationary (and hence doesn’t have a unit root), then Yt doesn’t contain a stochastic trend
Main methods for dealing with problem of spurious regression?
- Test for co-integration
2. Difference the data so it becomes stationary
- Benefit of co-integration (rather than differencing data) if possible, when dealing with problem of spurious regression?
- How do we do this?
1a. Co-integration allows us to see long-run relationship between X and Y
1b. Regressing on differences only allows short-run relationship
- Use error correction model
Initial (informal) indication of a stochastic trend?
- Fit a mean line through the data and see how often the series crosses the line
- If it doesn’t cross the line very often, this indicates data with stochastic trend
What is the implication for the standard Dickey-Fuller test if Yt is trend stationary?
- Test biased in favour of a unit root (only way model can fit trend is with unit root)
- High probability of type 1 error (rejecting null when it is true)
Under what assumption are the Dickey-Fuller critical values correct?
Errors serially uncorrelated
- If errors not serially uncorrelated, how can we ‘augment’ the Dickey-Fuller test?
- Explain how this works
- Augment DF test by adding lagged values of Y
2a. Want to ensure that any lagged differences with predictive power included in regression and not left in error term
2b. Need sufficient lags to ensure residuals are serially uncorrelated
When augmenting the DF test, what is the trade-off between using more/fewer lags?
- Too few lags - errors may be serially uncorrelated, meaning critical values are wrong
- Too many lags - larger standard error (less precise estimates) because observations and degrees of freedom lost when adding lags
How to decide how many lags to use in a DF test?
- Do sequence of F-tests,
2. Choose regression with the lowest information criterion
Why must we be cautious about accepting the null hypothesis in a DF unit root test?
- ADF test null hypothesis = series is non-stationary
- Accepting null hypothesis of unit root can be due to type 2 error (failing to reject the null when it is false)
- ADF test has low power to distinguish between unit roots and persistent but stationary alternatives
ADF test has low power to distinguish between ….. and …..
ADF test has low power to distinguish between unit roots and persistent but stationary alternatives
What can we NOT conclude if we fail to reject the null hypothesis in a unit root test?
Failure to reject null doesn’t mean the series does have a unit root (just insufficient evidence to conclude that it doesn’t)
What is a break?
Change in probability distribution of data (e.g. change in mean, variance etc)
Problems caused by breaks
- Destroy external validity of time series models
- Cause biased in-sample estimates of coefficients (OLS estimates ‘average value’, which won’t correspond to true effect in any period if there’s a break)
What is the F-test called when testing for a break?
Chow test
It may be difficult to distinguish multiple breaks from …..
It may be difficult to distinguish multiple breaks from stochastic trends
Remedies for breaks?
- If know/assume break date, estimate separate models in different sub-periods
2a. But break could be due to effect of outliers (e.g. outliers due to economic crisis in 1980, but in fact no break from 70s-90s)
2b. Could remove observations from sample, or model period w/outliers separately
- If break due to regime shift (e.g. change in inflation target from 1985), can model using dummy variables
- Examples of 2 cointegrated variables
2. Explain
- Short and long-run interest rates
(i) Economic theory - long rate = expected weighted sum of short rates over same time horizon
(ii) If short rate has stochastic trend, long rates must inherit that same trend (if this theory is correct)
(iii) i.e. long and short rates share common stochastic trend (they are co-integrated)
- If the law of one price holds, then exchange rates and price differentials must be cointegrated of order 1
What assumption is required for the identification of dynamic causal effects?
Random assignment of X (independent of potential outcomes), AKA EXOGENEITY
Under what assumption does OLS estimate the dynamic causal effect on Y of change in X?
X is EXOGENOUS
Distributed lag model assumptions
(1) X (regressors) exogenous
(2a) Y and X have stationary distributions
(2b) Observations (Yt, Xt) and (Yt-j, Xt-j) become independent as j gets large
(3) Y and X have non-zero finite 8th moments (E[Y^8]=0)
(4) No perfect multi-collinearity
Time series counter-part of ‘identically distributed’ part of usual iid assumption
X and Y have stationary distributions
Time series counter-part of ‘independently distributed’ part of usual iid assumption
(Yt, Xt) and (Yt-j, Xt-j) become independent as j gets large
What are the implications of the assumption for distributed lag models that X and Y have stationary distributions?
- Coefficients don’t change within sample (internal validity)
- Results can be extrapolated outside sample (external validity)
Intuition behind the assumption for distributed lag models that (Yt, Xt) and (Yt-j, Xt-j) become independent as j gets large
- We have separate experiments for different time periods (that are widely separated)
- Each new observations provides additional information (wouldn’t be case, for example, in extreme where data perfectly dependent over time)
Consequences of distributed lag model assumptions?
- OLS gives consistent estimators of B1, B2,… (dynamic multipliers)
- Sampling distribution of estimators of coefficients approximately normal in large samples
- But if errors serially correlated, then formula for variance and standard errors not the same as normal case of i.i.d data (instead use HAC SEs)
- Limitation of distributed lag model
2. Possible solutions
- Assumes all dynamic multipliers beyond lag r equal zero
2a. Solution - could add more lags
2b. But this means losing more observations at the beginning + degrees of freedom (which reduces precision of estimates)
- Solution - use ADL model
Verbally, explain what the following mean:
- Impact effect of change in X
- 1-period dynamic multiplier
- Effect of change in Xt on Yt, holding past Xt constant
2. Effect of change in Xt-1 on Yt, holding constant Xt, Xt-2, Xt-3,…
What does a 1-period dynamic multiplier mean?
Effect of change in Xt-1 on Yt, holding constant Xt, Xt-2, Xt-3,…
What does a 2-period dynamic multiplier mean?
Effect of change in Xt-2 on Yt, holding constant Xt, Xt-1, Xt-3,…
Example of a break
Change in inflation target
