TIME
SERIES (AUTOREGRESSIVE) MODELS
INTRODUCTION]
1. Causal premise: historical pattern of the dependent variable can be used to
forecast future values of the dependent variable under the assumption that past
influences will continue into the future.
2. Extrapolation of past time series into the
future (ex ante) can vary based upon
the mathematical form that most nearly described its pattern in the past (ex post).
3. Implications of extrapolation of historical
data for model selection:
a.
Time series models are best applied to an immediate or short term future
horizon.
b.
Time series models are most satisfactory when historical data patterns
are changing slowly and consistently (stationary series).
c.
Models can be simple and inexpensive (naive) to more complicated and
expensive (Box-Jenkins).
d.
Forecasts based upon past time patterns must be augmented by intuitive
judgment to determine other influences, especially as the time frame increases
to, say, six months.
PATTERN
IDENTIFICATION
1. The ACF of original data can be used to
determine if the data is stationary (no trend).
2. First differences removes a linear trend.
3. Second differences removes a quadratic
trend.
4. First differences of logarithms of data
removes a constant growth trend.
SIMPLE
TIME SERIES MODELS
1. The performance of a model is based upon its
ex post error terms rather than its mathematical sophistication.
2. Mean forecast for stationary data implies
that all other variation around its mean is either small or random.
3. A no change naive model allows for variation
in the forecast but without trend or seasonal variation.
4. Average change models adjust for historical
trends, but there will be a lag in turning points and all past values are
weighted equally.
5. Average percent changes give better forecast
for data with a constant growth rate but forecasts based upon more than one or
two months percent change will have a compounding effect of future forecasts
that must be avoided.
1. Table 8-2 shows the evaluation of historical
wage data presented in table 8-2. The
best model appears to be the average change model with n=2. Note that the evaluation of each model is
based upon its MAPE, MAD, mean error, and mean percent error. It does have a positive bias (over
forecasting wages on the average).
2. To evaluate the most recent performance the
last three data points may be removed from the data and the model
estimated. Table 8-3 shows that the
naive model outperforms the other two models on a simulated ex ante basis
because it is less likely to build up positive error terms over the three month
period. Because of its simplicity and
better more recent performance the naive model would be the best model.
3. We may decide to use the average change
model and the naive model as benchmarks against which more sophisticated models
could be evaluated. The model is always
updated each time a new data point is recorded.
4. Example 8-1 shows that we may combine the
two forecasts into one forecast by using a weighted average of the two forecast
values. The weighting scheme should
assign a higher weight to the forecast that generates the smallest error. A method of determining these weights is as
follows:
a.
Take each mean error as a percent of their combined mean error (ignoring
the signs).
b.
Determine the inverse of these percentages.
c.
Weight each forecast by this inverse to determine a combined forecast.
Autoregressive
Models
All autoregressive models
involve a determination of the order of the model (the number of lagged values
of the variable on the right-hand side of the equation) and the weights
assigned to each of the lagged values in the model.
Moving Average
Models
Simple Moving Average Models
1. Each data series may be converted into a new
series that is a moving average over any number of periods. This moving average smooths out
irregularities and captures cyclical influences if the data is stationary and
seasonally adjusted. Simple moving
average models have an order = n and weights = 1/n. Any value of n may be used, but
the higher the value of n the less the amount of variation in the
forecasts.
2. A forecast for the next period is the moving
average of the current period.
3. The value and bias of the error terms is
evaluation in determining the usefulness of the model or if an alternative
number of periods should be tried.
4. Table 14.2 compares the forecasts and error
measures for two alternative simple moving average forecasting models with n = 2
and n = 4. Clearly n = 4 is preferred
over n = 2 based upon the lower MAD.
Problems with simple moving
averages:
1. The forecast will lag turning points if it
captures them at all (oversmoothing for high values of n).
2.
Forecasts
will be unreliable (biased) when there is a strong trend in the variable.
3. Past observations are given the same
“weight.” This can be overcome with a
weighted moving average as shown in Table 4.2.
This model has decreasing weights with the fraction of n decreasing each
term but the sum of the weights equal to one.
Double Moving Average Models
1. Double moving average models correct for a
trend.
2. The original data series is smoothed with a
single moving average of order n, (M)
3. The new smoothed series is smothed again
with a second moving average of order n, (Md)
4. For the two new series the following
parameters are calculated for each time period beginning with the first period
when both M and Md are available:
a = 2M - Md
b = (2/(n-1)) (M - Md)
Predicted Y (t+T) = a + b T
Alternative method of
dealing with a trend applies simple moving average forecast of first differences of a data
series. The forecasted change can be
added to the last value to determine next period’s forecast.
Limitations of Moving
Average Models
1. May require lengthly time series, especially
if double moving average required.
2. Weights equal to 1/n are arbitrary and give
equal value to all past values.
3. The “trial and error” determination of the optimal
value of n is time consuming.
4.
Forecasts
are mechanistic and unreliable except for immediate time period forecasts.
Exponential
Smoothing
Simple exponential smoothing
forecast
1. Begin with an initial
smoothed value (often the initial value or an average of several recent values)
and an assumed smoothing constant that is a positive fraction.
2.
The smoothed series is updated by multiplying the most recent value
times the smoothing constant plus 1 minus the smoothing constant times the
previous smoothed value.
3.
The forecast for the next period is the smoothed value of the previous
period.
Double exponential smoothing
forecast (Brown model)
1. For data that is not
stationary a single exponential smoothing forecast will be biased. Double exponential smoothing is one method
of correcting for the trend in the data.
2.
Begin by determining an exponential smoothed series for the original
data based upon an assumed value of alpha and an initial value of S.
3.
Calculate an exponential smoothed series of the first smoothed series
using the same value of alpha and initial value for Sd equal to S.
4.
Calculate a equal to 2*S - Sd
5.
Calculate b equal to (alpha/1 - alpha)*(S - Sd)
6.
The forecast for the T period is a + b T.
Holt’s model for
nonstationary data
1. An alternative model for
adjusting for the trend in series Y uses two smoothing constants, alpha for the
average of the smoothed series and beta for the change in the smoothed series,
called the trend series.
2.
The average series is computed by assuming an initial value for the
series, A, (either the present value of Y or an average of recent values) and a
smoothing constant, alpha, to update the A series by multiplying alpha times
the most recent value of Y and adding one minus alpha times the sum of the
previous values of the A series and the T series.
3.
The trend series is computed by assuming an initial value of the trend
(an average of the change in several recent values of Y or zero if a large
number of observations) and updating this value by multiplying beta times the
change in A and adding one minus beta times the previous value of T.
4.
The forecast for the p period is the sum of the previous value of A plus
p times the previous value of T.
Eviews enables the forecaster
to choose among various exponential smoothing models with the command: SMOOTH