Fitting time series
simple time series models sometimes outperform regression models fitted to
nonstationary time series X and Y generally don't stay perfectly "in
synch" over long periods of time--i.e., they do not usually maintain
a perfectly linear relationship--even if they are causally related.
may be some "omitted variable", say Z, which could in principle
explain some of the discrepancy in the relationship between X and Y--but
this is not the only possibility. For example, the nature of the
relationship between X and Y may simply change over time.
Remember that if X and Y are nonstationary,
this means that we cannot necessarily assume their statistical properties (such
as their correlations with each other) are constant over time.
the variables X and Z change relatively slowly from period to period,
it is possible that the information X(t) and Z(t) contain with
respect to Y(t) is already contained in Y(t-1), Y(t-2), etc.--i.e., in Y's
own recent history--which time series models are able to utilize.
In other words, recent values of Y
might be good "proxies" not only for the effect of X but also for
the effects of any omitted variables.
and macroeconomic times series often have strong contemporaneous
correlations, but significant leading correlations--i.e.,
cross-correlations with other variables at positive lags--are often hard
to find. Thus, regression models may be better at predicting the present
than the future.
get the best of both worlds--regression and time series models:
1. Stationarize the variables (by
differencing, logging, deflating, or whatever) before fitting a regression
you can find transformations that render the variables stationary, then
you have greater assurance that the correlations between them will be
stable over time.
also implicitly brings the recent history of the variables into the
Example: instead of regressing Y on X, regress DIFF(Y) on DIFF(X)
regression equation is now: Ỹ(t) - Y(t-1) = a + b(X(t) - X(t-1))
is equivalent to : Ỹ(t) = Y(t-1) + a + bX(t) - bX(t-1)
that this brings the prior values of both X and Y into the prediction.
2. Use lagged
versions of the variables in the regression model.
allows varying amounts of recent history to be brought into the forecast
of independent variables is often necessary in order for the regression
model to be able to predict the future--i.e., to predict what will happen
in period t based on knowledge of what happened up to period t-1
Example: instead of regressing Y on X, regress Y on LAG(X,1) and
regression equation is now Ỹ(t) = a + bX(t-1) + cY(t-1)
3. It often helps to
do both--i.e., to stationarize the variables by differencing, then use
lags of the stationarized variables.
Example: regress DIFF(Y) on LAG(DIFF(X),1)
The regression equation is now: Ỹ(t)
- Y(t-1) = a + b(X(t-1) - X(t-2)) + c(Y(t-1) - Y(t-2))
...which is equivalent to Ỹ(t) = Y(t-1) + a +
b(X(t-1) - X(t-2)) + c(Y(t-1) - Y(t-2))
decide which combinations of lags or differences to try?
1. Determine which
transformations (if any) are needed to stationarize each variable by looking at
time series plots and autocorrelation plots
series plots of stationary variables should have a well-defined mean
and a relatively constant variance (i.e., no heteroscedasticity).
autocorrelations of a nonstationary variable will be strongly
positive and non-noisy-looking out to a high number of lags
(often 10 or more)--i.e., they will be "slow to decay"
autocorrelation plot of a stationary variable will usually decay
into "noise" and/or hit negative values within 3 or 4 lags.
the lag-1 autocorrelation is already negative, no more
differencing is needed. If the lag-1 autocorrelation is -0.5 or more
negative, the variable may have been "overdifferenced."
2. Look at
autocorrelations of the stationarized dependent variable (e.g., DIFF(Y)) to
determine whether one or more of its lags is likely to be helpful
mainly on what happens at the first few lags and (in the case of
seasonal data) what happens at the seasonal lag (e.g., at lag 12
for monthly data).
in the autocorrelation plot at peculiar lags (e.g. lag 5 or 7 in a
monthly time series) are probably mere "noise" caused by a
chance alignment of extreme values.
3. Look at cross-correlations
between the stationarized dependent variable (the "first" series)
and stationarized independent variables (the "second" series).
significant cross-correlation at a positive lag indicates that the
independent variable may be significant when lagged by that number of
example, if DIFF(X) is the second time series and a significant
cross-correlation is observed at lag 1, this suggests that LAG(DIFF(X),1)
might be a significant predictor of the dependent variable.
cross-correlation function, like the autocorrelation function, is
typically noisy. Cross-correlations at lags of 3 or more are
often merely accidental (except where seasonal effects are important).
Generally you should take seriously only the cross-correlations at lags 0,
1, and 2.
cross-correlation function, like the autocorrelation function, will
typically show a smooth pattern of significantly positive correlations out
to a high number of lags if the variables have not been properly
stationarized--but this doesn't mean much.
a regression model to differenced and/or lagged data:
1. Regress the
stationarized dependent variable on lags of itself and/or stationarized
independent variables as suggested by autocorrelation and cross-correlation
shows a significant autocorrelation at lags 1 and 2 but not at higher lags, and
DIFF(Y) shows a significant cross-correlation with DIFF(X) at lags 0 and 1.
regressing DIFF(Y) on LAG(DIFF(Y),1), LAG(DIFF(Y),2), and LAG(DIFF(X),1).
could also include DIFF(X) as indicated by the cross-correlation at lag 0,
but this would preclude being able to predict one period ahead.
your dependent variable is DIFF(Y), then the forecast reports and graphs
produced by your model will all be for differences of Y, not for Y
in its original units. The regression procedure does not automatically
"undifference" the output for you. (If you wish to print or plot
undifferenced forecasts, you will need to use spreadsheet commands either
in Statgraphics or Excel to add the predicted differences to the previous
actual values in order to get predictions for Y itself in each period.)
stock price data, on which you would probably use a DIFF(LOG( ))
transformation, obtaining results in differenced terms is not necessarily
bad: it is more interesting to plot the predicted returns rather
than predicted prices, so you can see if the predicted returns deviate
significantly from a constant.
2. You can use a
"stepwise" approach to model-fitting but beware of over-fitting the
data. Be cautious in choosing how many lags and how many different independent
variables to include at the beginning of the process.
Multiple Regression procedure includes automatic stepwise regression as a
right-mouse-button option. (The "forward" method is usually
Advanced Regression module contains an all-possible-regressions option
("Regression Model Selection" procedure)-- but beware!
3. It is especially
important to VALIDATE your model with hold-out data when selecting models by automatic
methods. Ideally you should withhold data during the model-selection process as
well as during the final testing of the model.
Multiple Regression procedure does not offer any options for validation,
Advanced Regression module includes a validation option in all its
procedures: use the "Select" field to hold out data--e.g.,
INDEX<81 to hold out all data after row 80.
good news: The
Forecasting procedure can be used to fit regression models to lagged and
differenced data and to validate them.
and lagging of the dependent variable can be performed by using ARIMA
example, to use DIFF(Y) as the dependent variable and include
LAG(DIFF(Y),1)) and LAG(DIFF(Y),2) as regressors, you would just use Y as
the input variable, then specify an ARIMA model with 1 nonseasonal
difference and set AR=2. Forecasts will be automatically
"undifferenced" in the output reports and graphs.
(Setting AR=2 means that you want 2 autoregressive
terms--i.e., the first two lags of the differenced dependent variable--included
in the forecasting equation.)
You can also, in principle, add regressors to the ARIMA model such as
LAG(DIFF(X),1) etc. However, the regression option gives a
"data error" message if there is more than 1 missing value at the
beginning of a regressor, which will be the case if the total number of lags or
differences is more than 1. (The regressor could either be lagged by one
period, or else differenced, but not both.)
- Workaround: use the "Generate Data" command to create
new columns on your data spreadsheet containing all lagged and/or
differenced variables that may be used as regressors, and assign them
short descriptive names. Then delete any rows at the beginning of the
file that contain missing values of these variables. In the
Forecasting procedure, specify the regressor variables by name
rather than using expressions that involve LAG or DIFF.