following is the original site for the Duke document on ARIMA models. It is the best I
have found in introducing you to what the ARIMA models do that connects to the
approach we have been using in class, particularly on assignment 1.
http://www.duke.edu/~rnau/411arim.htm
April 7, 2010
Introduction to ARIMA:
nonseasonal models
ARIMA(p,d,q)
ARIMA(0,1,0) = random walk
ARIMA(1,1,0) = differenced first-order autoregressive model
ARIMA(0,1,1) without constant = simple exponential smoothing
ARIMA(0,1,1) with constant = simple exponential smoothing with growth
ARIMA(0,2,1) or (0,2,2) without constant = linear exponential smoothing
A "mixed" model--ARIMA(1,1,1)
Spreadsheet implementation
ARIMA(p,d,q): ARIMA models are, in theory, the most general class of models for
forecasting a time series which can be stationarized by transformations such as
differencing and logging. In fact, the easiest way to think of ARIMA models is as finetuned versions of random-walk and random-trend models: the fine-tuning consists of
adding lags of the differenced series and/or lags of the forecast errors to the
prediction equation, as needed to remove any last traces of autocorrelation from the
forecast errors.
The acronym ARIMA stands for "Auto-Regressive Integrated Moving Average." Lags
of the differenced series appearing in the forecasting equation are called "autoregressive" terms, lags of the forecast errors are called "moving average" terms, and
a time series which needs to be differenced to be made stationary is said to be an
"integrated" version of a stationary series. Random-walk and random-trend
models, autoregressive models, and exponential smoothing models (i.e.,
exponential weighted moving averages) are all special cases of ARIMA
models.
A nonseasonal ARIMA model is classified as an "ARIMA(p,d,q)" model, where:

p is the number of autoregressive terms,
autoregression and autocorrelation are used interchangeably. In the
correlograms we used in class, when we were above .15 or below -.15
we were indentifying that autocorrelation existed. When such
autocorrelation existed in our “lag 1” model then p=1. If the
correlogram showed that a “lag 2” model also was above .15 or below .15 then p=2. So, in choosing our ARIMA model, we can make an
intelligent guess about “p” by looking at a correlogram.


d is the number of nonseasonal differences, and
q is the number of lagged forecast errors in the prediction equation.
To identify the appropriate ARIMA model for a time series, you begin by identifying
the order(s) of differencing needing to stationarize the series and remove the gross
features of seasonality, perhaps in conjunction with a variance-stabilizing
“variance stabilizing”- remember how the first differences got
bigger through time? We used percentage changes to get rid of
that problem. But by choosing the correct parameters (p,d,q) on
an ARIMA model we can take care of this problem
transformation such as logging or deflating. If you stop at this point and predict that
the differenced series is constant, you have merely fitted a random walk
“random walk model”. The original “random walk” model was
illustrated by envisioning a drunk at a light post in the middle of
the night. The “random walk” describes the path that drunk
would be likely to make around the light post before collapsing.
If the “random walk” does not have a “constant term”, the drunk
will collapse on the lamp post. If the random walk is
“stationary” the drunk will collapse at the same spot every time
the drunk gets up, wanders around and collapses again. (sorry
for the graphic nature of the illustration).
or random trend model. (Recall that the random walk model predicts the first
difference of the series to be constant, the seasonal random walk model predicts the
seasonal difference to be constant, and the seasonal random trend model predicts
the first difference of the seasonal difference to be constant--usually zero.) However,
the best random walk or random trend model may still have autocorrelated errors,
suggesting that additional factors of some kind are needed in the prediction
equation.
ARIMA(0,1,0) = random walk: In models we have studied previously, we have
encountered two strategies for eliminating autocorrelation in forecast errors. One
approach, which we first used in regression analysis, was the addition of lags of the
stationarized series. For example, suppose we initially fit the random-walk-withgrowth model to the time series Y. The prediction equation for this model can be
written as:
(1) The “Naïve” Model=Regressing a variable on itself= ARIMA(0,1,0)
...where the constant term (here denoted by "mu") is the average difference in Y.
in other words “” indicates how far from the lamppost the drunk
collapses
This can be considered as a degenerate regression model in which DIFF(Y) is the
dependent variable and there are no independent variables other than the constant
term. Since it includes (only) a nonseasonal difference and a constant term, it is
classified as an "ARIMA(0,1,0) model with constant." Of course, the random walk
without growth would be just an ARIMA(0,1,0) model without constant.
without constant, collapsing means hitting the lamppost, but
with constant collapsing happens out there in the dark
somewhere.
ARIMA(1,1,0) = differenced first-order autoregressive model: If the errors of
the random walk model are autocorrelated, perhaps the problem can be fixed by
adding one lag of the dependent variable to the prediction equation--i.e., by
regressing DIFF(Y) on itself lagged by one period. This would yield the following
prediction equation:
(2) Regressing Change (first Difference) on one period lagged change=
ARIMA(1,1,0)
Everything that comes after the “” is simply a way of saying the
“change” from one period earlier.
which can be rearranged to
This is a first-order autoregressive, or "AR(1)", model with one order of nonseasonal
differencing and a constant term--i.e., an "ARIMA(1,1,0) model with constant." Here,
the constant term is denoted by "mu" and the autoregressive coefficient is denoted
by "phi", in keeping with the terminology for ARIMA models popularized by Box and
Jenkins. (In the output of the Forecasting procedure in Statgraphics, this coefficient
is simply denoted as the AR(1) coefficient.)
ARIMA(0,1,1) without constant = simple exponential smoothing: Another
strategy for correcting autocorrelated errors in a random walk model is suggested by
the simple exponential smoothing model. Recall that for some nonstationary time
series (e.g., one that exhibits noisy fluctuations around a slowly-varying mean), the
random walk model does not perform as well as a moving average of past values. In
other words, rather than taking the most recent observation as the forecast of the
next observation, it is better to use an average of the last few observations in order
to filter out the noise and more accurately estimate the local mean. The simple
exponential smoothing model uses an exponentially weighted moving average of
past values to achieve this effect.
We saw this in the chapter on weighted moving averages.
The prediction equation for the simple exponential smoothing model can be written
in a number of mathematically equivalent ways, one of which is:
...where e(t-1) denotes the error at period t-1. Note that this resembles the
prediction equation for the ARIMA(1,1,0) model, except that instead of a multiple of
the lagged difference it includes a multiple of the lagged forecast error. (It also does
not include a constant term--yet.) The coefficient of the lagged forecast error is
denoted by the Greek letter "theta" (again following Box and Jenkins) and it is
conventionally written with a negative sign for reasons of mathematical symmetry.
"Theta" in this equation corresponds to the quantity "1-minus-alpha" in the
exponential smoothing formulas we studied earlier.
When a lagged forecast error is included in the prediction equation as shown above,
it is referred to as a "moving average" (MA) term. The simple exponential smoothing
model is therefore a first-order moving average ("MA(1)") model with one order of
nonseasonal differencing and no constant term --i.e., an "ARIMA(0,1,1) model
without constant." This means that in Statgraphics (or any other statistical software
that supports ARIMA models) you can actually fit a simple exponential smoothing by
specifying it as an ARIMA(0,1,1) model without constant, and the estimated MA(1)
coefficient corresponds to "1-minus-alpha" in the SES formula.
ARIMA(0,1,1) with constant = simple exponential smoothing with growth:
By implementing the SES model as an ARIMA model, you actually gain some
flexibility. First of all, the estimated MA(1) coefficient is allowed to be negative: this
corresponds to a smoothing factor larger than 1 in an SES model, which is usually
not allowed by the SES model-fitting procedure. Second, you have the option of
including a constant term in the ARIMA model if you wish, in order to estimate an
average non-zero trend. The ARIMA(0,1,1) model with constant has the prediction
equation:
The one-period-ahead forecasts from this model are qualitatively similar to those of
the SES model, except that the trajectory of the long-term forecasts is typically a
sloping line (whose slope is equal to mu) rather than a horizontal line.
ARIMA(0,2,1) or (0,2,2) without constant = linear exponential smoothing:
Linear exponential smoothing models are ARIMA models which use two nonseasonal
differences in conjunction with MA terms. The second difference of a series Y is not
simply the difference between Y and itself lagged by two periods, but rather it is the
first difference of the first difference--i.e., the change-in-the-change of Y at period t.
Thus, the second difference of Y at period t is equal to (Y(t)-Y(t-1)) - (Y(t1)-Y(t-2)) = Y(t) - 2Y(t-1) + Y(t-2). A second difference of a discrete function is
analogous to a second derivative of a continuous function: it measures the
"acceleration" or "curvature" in the function at a given point in time.
The ARIMA(0,2,2) model without constant predicts that the second difference of the
series equals a linear function of the last two forecast errors:
which can be rearranged as:
where theta-1 and theta-2 are the MA(1) and MA(2) coefficients. This is essentially
the same as Brown's linear exponential smoothing model, with the MA(1) coefficient
corresponding to the quantity 2*(1-alpha) in the LES model. To see this connection,
recall that forecasting equation for the LES model is:
Upon comparing terms, we see that the MA(1) coefficient corresponds to the
quantity 2*(1-alpha) and the MA(2) coefficient corresponds to the quantity -(1alpha)^2 (i.e., "minus (1-alpha) squared"). If alpha is larger than 0.7, the
corresponding MA(2) term would be less than 0.09, which might not be significantly
different from zero, in which case an ARIMA(0,2,1) model probably would be
identified.
A "mixed" model--ARIMA(1,1,1): The features of autoregressive and moving
average models can be "mixed" in the same model. For example, an ARIMA(1,1,1)
model with constant would have the prediction equation:
Normally, though, we will try to stick to "unmixed" models with either only-AR or
only-MA terms, because including both kinds of terms in the same model sometimes
leads to overfitting of the data and non-uniqueness of the coefficients.
Spreadsheet implementation: ARIMA models such as those described above are
easy to implement on a spreadsheet. The prediction equation is simply a linear
equation that refers to past values of original time series and past values of the
errors. Thus, you can set up an ARIMA forecasting spreadsheet by storing the data in
column A, the forecasting formula in column B, and the errors (data minus forecasts)
in column C. The forecasting formula in a typical cell in column B would simply be a
linear expression referring to values in preceding rows of columns A and C, multiplied
by the appropriate AR or MA coefficients stored in cells elsewhere on the
spreadsheet.
Go to next topic: Identifying the order of differencing
Identifying the order of differencing
The first (and most important) step in fitting an ARIMA model is the determination of
the order of differencing needed to stationarize the series. Normally, the
correct amount of differencing is the lowest order of
differencing that yields a time series which fluctuates around a
well-defined mean value and whose autocorrelation function
(ACF) plot decays fairly rapidly to zero, either from above or below.
the ”autocorrelation function (ACF) plot” is what we have been
calling a correlogram.
If the series still exhibits a long-term trend, or otherwise lacks a tendency to return
to its mean value, or if its autocorrelations are positive out to a high number of lags
(e.g., 10 or more), then it needs a higher order of differencing.
for example if the correlogram has a bunch of lags above .15 for a
“change” model (ARIMA(0,1,0)), then go to a “change of the
change” model (ARIMA(0,2,0)). The correlogram should
deteriorate more rapidly.
We will designate this as our "first rule of identifying ARIMA models" :

Rule 1: If the series has positive autocorrelations out to a high
number of lags, then it probably needs a higher order of differencing.
Differencing tends to introduce negative correlation: if the series initially shows
strong positive autocorrelation, then a nonseasonal difference will reduce the
autocorrelation and perhaps even drive the lag-1 autocorrelation to a negative value.
If you apply a second nonseasonal difference (which is occasionally necessary), the
lag-1 autocorrelation will be driven even further in the negative direction.
If the lag-1 autocorrelation is zero or even negative, then the series does not need
further differencing. You should resist the urge to difference it anyway just because
you don't see any pattern in the autocorrelations! One of the most common errors in
ARIMA modeling is to "overdifference" the series and end up adding extra AR or MA
terms to undo the damage. If the lag-1 autocorrelation is more negative than -0.5
(and theoretically a negative lag-1 autocorrelation should never be greater than 0.5
in magnitude), this may mean the series has been overdifferenced. The time series
plot of an overdifferenced series may look quite random at first glance, but if you
look closer you will see a pattern of excessive changes in sign from one observation
to the next--i.e., up-down-up-down, etc. :

Rule 2: If the lag-1 autocorrelation is zero or negative, or the
autocorrelations are all small and patternless, then the series does
not need a higher order of differencing. If the lag-1 autocorrelation is
-0.5 or more negative, the series may be overdifferenced. BEWARE
OF OVERDIFFERENCING!!
Another symptom of possible overdifferencing is an increase in the standard
deviation, rather than a reduction, when the order of differencing is increased. This
becomes our third rule:

Rule 3: The optimal order of differencing is often the order of
differencing at which the standard deviation is lowest.
In the Forecasting procedure in Statgraphics, you can find the order of differencing
that minimizes the standard deviation by fitting ARIMA models with various orders of
differencing and no coefficients other than a constant. For example, if you fit an
ARIMA(0,0,0) model with constant, an ARIMA(0,1,0) model with constant, and an
ARIMA(0,2,0) model with constant, then the RMSE's will be equal to the standard
deviations of the original series with 0, 1, and 2 orders of nonseasonal differencing,
respectively. The first two rules do not always unambiguously determine the
"correct" order of differencing. We will see later that "mild underdifferencing" can be
compensated for by adding AR terms to the model, while "mild overdifferencing" can
be compensated for by adding MA terms instead. In some cases, there may be two
different models which fit the data almost equally well: a model that uses 0 or 1
order of differencing together with AR terms, versus a model that uses the next
higher order of differencing together with MA terms. In trying to choose between two
such models that use different orders of differencing, you may need to ask what
assumption you are most comfortable making about the degree of nonstationarity in
the original series--i.e., the extent to which it does or doesn't have fixed mean
and/or a constant average trend.

Rule 4: A model with no orders of differencing assumes that the
original series is stationary (mean-reverting). A model with one order
of differencing assumes that the original series has a constant
average trend (e.g. a random walk or SES-type model, with or
without growth). A model with two orders of total differencing
assumes that the original series has a time-varying trend (e.g. a
random trend or LES-type model).
Another consideration in determining the order of differencing is the role played by
the CONSTANT term in the model--if one is included. The constant represents the
mean of the series if no differencing is performed, it represents the average trend in
the series if one order of differencing is used, and it represents that average trend-
in-the-trend (i.e., curvature) if there are two orders of differencing. We generally do
not assume that there are trends-in-trends, so the constant is usually removed from
models with two orders of differencing. In a model with one order of differencing, the
constant may or may not be included, depending on whether we do or do not want
to allow for an average trend. Hence we have:

Rule 5: A model with no orders of differencing normally includes a
constant term (which represents the mean of the series). A model
with two orders of total differencing normally does not include a
constant term. In a model with one order of total differencing, a
constant term should be included if the series has a non-zero average
trend.
An example: Consider the UNITS series in the TSDATA sample data file that comes
with SGWIN. (This is a nonseasonal time series consisting of unit sales data.) First
let's look at the series with zero orders of differencing--i.e., the original time series.
There are many ways we could obtain plots of this series, but let's do so by
specifying an ARIMA(0,0,0) model with constant--i.e., an ARIMA model with no
differencing and no AR or MA terms, only a constant term. This is just the "mean"
model under another name, and the time series plot of the residuals is therefore just
a plot of deviations from the mean:
The autocorrelation function (ACF) plot shows a very slow, linear decay pattern
which is typical of a nonstationary time series:
This is just the correlogram you have been computing for a
variable upon itself.
The RMSE (which is just the standard deviation of the residuals in a constant-only
model) shows up as the "estimated white noise standard deviation" in the Analysis
Summary:
Forecast model selected: ARIMA(0,0,0) with constant
ARIMA Model Summary
Parameter
Estimate
Stnd. Error
t
P-value
----------------------------------------------------------------------Mean
222.738
1.60294
138.956
0.000
Constant
222.738
----------------------------------------------------------------------Backforecasting: yes
Estimated white noise variance = 308.329 with 149 degrees of freedom
Estimated white noise standard deviation = 17.5593
Number of iterations: 3
Clearly at least one order of differencing is needed to stationarize this series. After
taking one nonseasonal difference--i.e., fitting an ARIMA(0,1,0) model with constant-the residuals look like this:
They are trying to pick how many ”p,d,q” features you need to
prevent autocorrelation from being a problem while improving the
forecasting capability of the model. They are showing you how it’s
done by taking you through this example. note the use of the pvalue to show how well the model does in fitting the data. The
number of iterations refers to the number of times the model has
to go back to estimate the parameters to the equation being used
to forecast the series.
Notice that the series appears approximately stationary with no long-term trend: it
exhibits a definite tendency to return to its mean, albeit a somewhat lazy one. The
ACF plot confirms a slight amount of positive autocorrelation:
In other words, by using a “change” (first difference) model they
have eliminated a lot of the significant autocorrelation- but not for
the first three lags!!! They’re still above. 15.
The standard deviation has been dramatically reduced from 17.5593 to 2.38 as
shown in the Analysis Summary:
Forecast model selected: ARIMA(0,1,0) with constant
ARIMA Model Summary
Parameter
Estimate
Stnd. Error
t
P-value
----------------------------------------------------------------------Mean
0.50095
0.141512
3.53999
0.000535
Constant
0.50095
----------------------------------------------------------------------Backforecasting: yes
Estimated white noise variance = 2.38304 with 148 degrees of freedom
Estimated white noise standard deviation = 1.54371
Number of iterations: 2
Is the series stationary at this point, or is another difference needed? Because the
trend has been completely eliminated and the amount of autocorrelation which
remains is small, it appears as though the series may be satisfactorily stationary. If
we try a second nonseasonal difference--i.e., an ARIMA(0,2,0) model--just to see
what the effect is, we obtain the following time series plot:
If you look closely, you will notice the signs of overdifferencing--i.e., a pattern of
changes of sign from one observation to the next. This is confirmed by the ACF plot,
which now has a negative spike at lag 1 that is close to 0.5 in magnitude:
Is the series now overdifferenced? Apparently so, because the standard deviation
has actually increased from 1.54371 to 1.81266:
Forecast model selected: ARIMA(0,2,0) with constant
ARIMA Model Summary
Parameter
Estimate
Stnd. Error
t
P-value
----------------------------------------------------------------------Mean
0.000782562
0.166869
0.00468969 0.996265
Constant
0.000782562
----------------------------------------------------------------------Backforecasting: yes
Estimated white noise variance = 3.28573 with 147 degrees of freedom
Estimated white noise standard deviation = 1.81266
Number of iterations: 1
Thus, it appears that we should start by taking a single nonseasonal difference.
However, this is not the last word on the subject: we may find when we add AR or
MA terms that a model with another order of differencing works a little better. Or, we
may conclude that the properties of the long-term forecasts are more intuitively
reasonable with another order of differencing (more about this later). But for now,
we will go with one order of nonseasonal differencing.
Go to next topic: Identifying the orders of AR or MA terms.
Identifying the numbers of AR or MA terms
ACF and PACF plots
AR and MA signatures
A model for the UNITS series--ARIMA(2,1,0)
Mean versus constant
Alternative model for the UNITS series--ARIMA(0,2,1)
Which model should we choose?
Mixed models
Unit roots
ACF and PACF plots: After a time series has been stationarized by differencing, the
next step in fitting an ARIMA model is to determine whether AR or MA terms are
needed to correct any autocorrelation that remains in the differenced series. Of
course, with software like Statgraphics, you could just try some different
combinations of terms and see what works best. But there is a more systematic way
to do this. By looking at the autocorrelation function (ACF) and partial
autocorrelation (PACF) plots of the differenced series, you can tentatively identify
the numbers of AR and/or MA terms that are needed. You are already familiar with
the ACF plot: it is merely a bar chart of the coefficients of correlation between a time
series and lags of itself. The PACF plot is a plot of the partial correlation coefficients
between the series and lags of itself.
In general, the "partial" correlation between two variables is the amount of
correlation between them which is not explained by their mutual correlations with a
specified set of other variables. For example, if we are regressing a variable Y on
other variables X1, X2, and X3, the partial correlation between Y and X3 is the
amount of correlation between Y and X3 that is not explained by their common
correlations with X1 and X2. This partial correlation can be computed as the square
root of the reduction in variance that is achieved by adding X3 to the regression of Y
on X1 and X2.
A partial autocorrelation is the amount of correlation between a variable and a lag of
itself that is not explained by correlations at all lower-order-lags. The autocorrelation
of a time series Y at lag 1 is the coefficient of correlation between Y(t) and Y(t-1),
which is presumably also the correlation between Y(t-1) and Y(t-2). But if Y(t) is
correlated with Y(t-1), and Y(t-1) is equally correlated with Y(t-2), then we should
also expect to find correlation between Y(t) and Y(t-2). (In fact, the amount of
correlation we should expect at lag 2 is precisely the square of the lag-1 correlation.)
Thus, the correlation at lag 1 "propagates" to lag 2 and presumably to higher-order
lags. The partial autocorrelation at lag 2 is therefore the difference between the
actual correlation at lag 2 and the expected correlation due to the propagation of
correlation at lag 1.
Here is the autocorrelation function (ACF) of the UNITS series, before any
differencing is performed:
The autocorrelations are significant for a large number of lags--but perhaps the
autocorrelations at lags 2 and above are merely due to the propagation of the
autocorrelation at lag 1. This is confirmed by the PACF plot:
Note that the PACF plot has a significant spike only at lag 1, meaning that all the
higher-order autocorrelations are effectively explained by the lag-1 autocorrelation.
The PACF plot is just like the correlogram EXCEPT that
correlation shown in the second lag is NET of what was explained
in the first lag. The correlation shown in the the third lage is NET
of what was explained the previous two lags. By using multiple
regression with all of the lags you are getting these net
correlations- some of you did this on quiz08.
The partial autocorrelations at all lags can be computed by fitting a succession of
autoregressive models with increasing numbers of lags. In particular, the partial
autocorrelation at lag k is equal to the estimated AR(k) coefficient in an
autoregressive model with k terms--i.e., a multiple regression model in which Y is
regressed on LAG(Y,1), LAG(Y,2), etc., up to LAG(Y,k). Thus, by mere inspection of
the PACF you can determine how many AR terms you need to use to explain the
autocorrelation pattern in a time series: if the partial autocorrelation is significant at
lag k and not significant at any higher order lags--i.e., if the PACF "cuts off" at lag k-then this suggests that you should try fitting an autoregressive model of order k
The PACF of the UNITS series provides an extreme example of the cut-off
phenomenon: it has a very large spike at lag 1 and no other significant spikes,
indicating that in the absence of differencing an AR(1) model should be used.
However, the AR(1) term in this model will turn out to be equivalent to a first
difference, because the estimated AR(1) coefficient (which is the height of the PACF
spike at lag 1) will be almost exactly equal to 1. Now, the forecasting equation for an
AR(1) model for a series Y with no orders of differencing is:
If the AR(1) coefficient ("phi") in this equation is equal to 1, it is equivalent to
predicting that the first difference of Y is constant--i.e., it is equivalent to the
equation of the random walk model with growth:
The PACF of the UNITS series is telling us that, if we don't difference it, then we
should fit an AR(1) model which will turn out to be equivalent to taking a first
difference. In other words, it is telling us that UNITS really needs an order of
differencing to be stationarized.
In their case, here, the naïve model works!!!!!
AR and MA signatures: If the PACF displays a sharp cutoff while the ACF decays
more slowly (i.e., has significant spikes at higher lags), we say that the stationarized
series displays an "AR signature," meaning that the autocorrelation pattern can be
explained more easily by adding AR terms than by adding MA terms. You will
probably find that an AR signature is commonly associated with positive
autocorrelation at lag 1--i.e., it tends to arise in series which are slightly
underdifferenced. The reason for this is that an AR term can act like a "partial
difference" in the forecasting equation. For example, in an AR(1) model, the AR
term acts like a first difference if the autoregressive coefficient is equal to 1, it does
nothing if the autoregressive coefficient is zero, and it acts like a partial difference if
the coefficient is between 0 and 1. So, if the series is slightly underdifferenced--i.e. if
the nonstationary pattern of positive autocorrelation has not completely been
eliminated, it will "ask for" a partial difference by displaying an AR signature. Hence,
we have the following rule of thumb for determining when to add AR terms:

Rule 6: If the PACF of the differenced series displays a sharp cutoff
and/or the lag-1 autocorrelation is positive--i.e., if the series appears
slightly "underdifferenced"--then consider adding an AR term to the
model. The lag at which the PACF cuts off is the indicated number of
AR terms.
In principle, any autocorrelation pattern can be removed from a stationarized series
by adding enough autoregressive terms (lags of the stationarized series) to the
forecasting equation, and the PACF tells you how many such terms are likely be
needed. However, this is not always the simplest way to explain a given pattern of
autocorrelation: sometimes it is more efficient to add MA terms (lags of the forecast
errors) instead. The autocorrelation function (ACF) plays the same role for MA terms
that the PACF plays for AR terms--that is, the ACF tells you how many MA terms are
likely to be needed to remove the remaining autocorrelation from the differenced
series. If the autocorrelation is significant at lag k but not at any higher lags--i.e., if
the ACF "cuts off" at lag k--this indicates that exactly k MA terms should be used in
the forecasting equation. In the latter case, we say that the stationarized series
displays an "MA signature," meaning that the autocorrelation pattern can be
explained more easily by adding MA terms than by adding AR terms.
An MA signature is commonly associated with negative autocorrelation at lag 1--i.e.,
it tends to arise in series which are slightly overdifferenced. The reason for this is
that an MA term can "partially cancel" an order of differencing in the
forecasting equation. To see this, recall that an ARIMA(0,1,1) model without
constant is equivalent to a Simple Exponential Smoothing model. The forecasting
equation for this model is
where the MA(1) coefficient "theta" corresponds to the quantity "1-alpha" in the SES
model. If the MA(1) coefficient is equal to 1, this corresponds to an SES model with
alpha=0, which is just a CONSTANT model because the forecast is never updated.
This means that when the MA(1) coefficient is equal to 1, it is actually cancelling out
the differencing operation that ordinarily enables the SES forecast to re-anchor itself
on the last observation. On the other hand, if the moving-average coefficient is equal
to 0, this model reduces to a random walk model--i.e., it leaves the differencing
operation alone. So, if the MA(1) coefficient is something greater than 0, it is as if we
are partially cancelling an order of differencing . If the series is already slightly
overdifferenced--i.e., if negative autocorrelation has been introduced--then it will
"ask for" a difference to be partly cancelled by displaying an MA signature. (A lot of
arm-waving is going on here! A more rigorous explanation of this effect is found in
the "Mathematics of ARIMA Models" handout.) Hence the following additional rule of
thumb:

Rule 7: If the ACF of the differenced series displays a sharp cutoff
and/or the lag-1 autocorrelation is negative--i.e., if the series
appears slightly "overdifferenced"--then consider adding an MA term
to the model. The lag at which the ACF cuts off is the indicated
number of MA terms.
A model for the UNITS series--ARIMA(2,1,0): Previously we determined that the
UNITS series needed (at least) one order of nonseasonal differencing to be
stationarized. After taking one nonseasonal difference--i.e., fitting an ARIMA(0,1,0)
model with constant--the ACF and PACF plots look like this:
Notice that (a) the correlation at lag 1 is significant and positive, and (b) the PACF
shows a sharper "cutoff" than the ACF. In particular, the PACF has only two
significant spikes, while the ACF has four. Thus, according to Rule 7 above, the
differenced series displays an AR(2) signature. If we therefore set the order of the
AR term to 2--i.e., fit an ARIMA(2,1,0) model--we obtain the following ACF and PACF
plots for the residuals:
The autocorrelation at the crucial lags--namely lags 1 and 2--has been eliminated,
and there is no discernible pattern in higher-order lags. The time series plot of the
residuals shows a slightly worrisome tendency to wander away from the mean:
However, the analysis summary report shows that the model nonetheless performs
quite well in the validation period, both AR coefficients are significantly different from
zero, and the standard deviation of the residuals has been reduced from 1.54371 to
1.4215 (nearly 10%) by the addition of the AR terms. Furthermore, there is no sign
of a "unit root" because the sum of the AR coefficients (0.252254+0.195572) is not
close to 1. (Unit roots are discussed on more detail below.) On the whole, this
appears to be a good model.
Analysis Summary
Data variable: units
Number of observations = 150
Start index = 1/80
Sampling interval = 1.0 month(s)
Length of seasonality = 12
Forecast Summary
---------------Nonseasonal differencing of order: 1
Forecast model selected: ARIMA(2,1,0) with constant
Number of forecasts generated: 24
Number of periods withheld for validation: 30
Estimation
Validation
Statistic
Period
Period
-------------------------------------------MSE
2.10757
0.757818
MAE
1.11389
0.726546
MAPE
0.500834
0.280088
ME
0.0197748
-0.200813
MPE
0.0046833
-0.0778831
ARIMA Model Summary
Parameter
Estimate
Stnd. Error
t
P-value
----------------------------------------------------------------------AR(1)
0.252254
0.0915209
2.756240
0.006593
AR(2)
0.195572
0.0916343
2.13426
0.034492
Mean
0.467566
0.240178
1.94675
0.053485
Constant
0.258178
--------------------------------------------------------------------------Backforecasting: yes
Estimated white noise variance = 2.10875 with 146 degrees of freedom
Estimated white noise standard deviation = 1.45215
Number of iterations: 1
The (untransformed) forecasts for the model show a linear upward trend projected
into the future:
The trend in the long-term forecasts is due to fact that the model includes one
nonseasonal difference and a constant term: this model is basically a random walk
with growth (NF#1) fine-tuned by the addition of two autoregressive terms--i.e., two
lags of the differenced series. The slope of the long-term forecasts (i.e., the average
increase from one period to another) is equal to the mean term in the model
summary (0.467566). The forecasting equation is:
where "mu" is the constant term in the model summary (0.258178), "phi-1" is the
AR(1) coefficient (0.25224) and "phi-2" is the AR(2) coefficient (0.195572).
Mean versus constant: In general, the "mean" term in the output of an ARIMA
model refers to the mean of the differenced series (i.e., the average trend if the
order of differencing is equal to 1), whereas the "constant" is the constant term that
appears on the right-hand-side of the forecasting equation. The mean and constant
terms are related by the equation:
CONSTANT = MEAN*(1 minus the sum of the AR coefficients).
In this case, we have 0.258178 = 0.467566*(1 - 0.25224 - 0.195572)
Alternative model for the UNITS series--ARIMA(0,2,1): Recall that when we
began to analyze the UNITS series, we were not entirely sure of the correct order of
differencing to use. One order of nonseasonal differencing yielded the lowest
standard deviation (and a pattern of mild positive autocorrelation), while two orders
of nonseasonal differencing yielded a more stationary-looking time series plot (but
with rather strong negative autocorrelation). Here are both the ACF and PACF of the
series with two nonseasonal differences:
The single negative spike at lag 1 in the ACF is an MA(1) signature, according to Rule
8 above. Thus, if we were to use 2 nonseasonal differences, we would also want to
include an MA(1) term, yielding an ARIMA(0,2,1) model. According to Rule 5, we
would also want to suppress the constant term. Here, then, are the results of fitting
an ARIMA(0,2,1) model without constant:
Analysis Summary
Data variable: units
Number of observations = 150
Start index = 1/80
Sampling interval = 1.0 month(s)
Forecast Summary
---------------Nonseasonal differencing of order: 2
Forecast model selected: ARIMA(0,2,1)
Number of forecasts generated: 24
Number of periods withheld for validation: 30
Estimation
Validation
Statistic
Period
Period
-------------------------------------------MSE
2.13793
0.856734
MAE
1.15376
0.771561
MAPE
0.518221
0.297298
ME
0.0267768
-0.038966
MPE
0.017097
-0.0148876
ARIMA Model Summary
Parameter
Estimate
Stnd. Error
t
P-value
----------------------------------------------------------------------MA(1)
0.75856
0.0607947
12.4774
.000000
----------------------------------------------------------------------Backforecasting: yes
Estimated white noise variance = 2.1404 with 147 degrees of freedom
Estimated white noise standard deviation = 1.46301
Number of iterations: 4
Notice that the estimated white noise standard deviation (RMSE) is only very slightly
higher for this model than the previous one (1.46301 here versus 1.45215
previously). The forecasting equation for this model is:
where theta-1 is the MA(1) coefficients. Recall that this is similar to a Linear
Exponential Smoothing model, with the MA(1) coefficient corresponding to the
quantity 2*(1-alpha) in the LES model. The MA(1) coefficient of 0.76 in this model
suggests that an LES model with alpha in the vicinity of 0.72 would fit about equally
well. Actually, when an LES model is fitted to the same data, the optimal value of
alpha turns out to be around 0.61, which is not too far off. Here is a model
comparison report that shows the results of fitting the ARIMA(2,1,0) model with
constant, the ARIMA(0,2,1) model without constant, and the LES model:
Model Comparison
---------------Data variable: units
Number of observations = 150
Start index = 1/80
Sampling interval = 1.0 month(s)
Number of periods withheld for validation: 30
Models
-----(A) ARIMA(2,1,0) with constant
(B) ARIMA(0,2,1)
(C) Brown's linear exp. smoothing with alpha = 0.6067
Estimation Period
Model MSE
MAE
MAPE
ME
MPE
----------------------------------------------------------------------(A)
2.10757
1.11389
0.500834
0.0197748
0.0046833
(B)
2.13793
1.15376
0.518221
0.0267768
0.017097
(C)
2.18056
1.14308
0.514482
0.000765696 0.0041274
Model RMSE
RUNS RUNM AUTO MEAN VAR
----------------------------------------------(A)
1.45175
OK
OK
OK
OK
OK
(B)
1.46217
OK
OK
OK
OK
OK
(C)
1.47667
OK
OK
OK
OK
OK
Validation Period
Model MSE
MAE
MAPE
ME
MPE
----------------------------------------------------------------------(A)
0.757818
0.726546
0.280088
-0.200813
-0.0778831
(B)
0.856734
0.771561
0.297298
-0.038966
-0.0148876
(C)
0.882234
0.796109
0.306593
-0.014557
-0.00527448
The three models perform nearly identically in the estimation period, and the
ARIMA(2,1,0) model with constant appears slightly better than the other two in the
validation period. On the basis of these statistical results alone, it would be hard to
choose among the three models. However, if we plot the long-term forecasts made
by the ARIMA(0,2,1) model without constant (which are essentially the same as
those of the LES model), we see a significant difference from those of the earlier
model:
The forecasts have somewhat less of an upward trend than those of the earlier
model--because the local trend near the end of the series is slightly less than the
average trend over the whole series--but the confidence intervals widen much more
rapidly. The model with two orders of differencing assumes that the trend in the
series is time-varying, hence it considers the distant future to be much more
uncertain than does the model with only one order of differencing.
Which model should we choose? That depends on the assumptions we are
comfortable making with respect to the constancy of the trend in the data. The
model with only one order of differencing assumes a constant average trend--it is
essentially a fine-tuned random walk model with growth--and it therefore makes
relatively conservative trend projections. It is also fairly optimistic about the
accuracy with which it can forecast more than one period ahead. The model with two
orders of differencing assumes a time-varying local trend--it is essentially a linear
exponential smoothing model--and its trend projections are somewhat more fickle.
As a general rule in this kind of situation, I would recommend choosing the model
with the lower order of differencing, other things being roughly equal. In practice,
random-walk or simple-exponential-smoothing models often seem to work better
than linear exponential smoothing models.
Mixed models: In most cases, the best model turns out a model that uses either
only AR terms or only MA terms, although in some cases a "mixed" model with both
AR and MA terms may provide the best fit to the data. However, care must be
exercised when fitting mixed models. It is possible for an AR term and an MA term to
cancel each other's effects, even though both may appear significant in the model
(as judged by the t-statistics of their coefficients). Thus, for example, suppose that
the "correct" model for a time series is an ARIMA(0,1,1) model, but instead you fit
an ARIMA(1,1,2) model--i.e., you include one additional AR term and one additional
MA term. Then the additional terms may end up appearing significant in the model,
but internally they may be merely working against each other. The resulting
parameter estimates may be ambiguous, and the parameter estimation process may
take very many (e.g., more than 10) iterations to converge. Hence:

Rule 8: It is possible for an AR term and an MA term to cancel each
other's effects, so if a mixed AR-MA model seems to fit the data, also
try a model with one fewer AR term and one fewer MA term-particularly if the parameter estimates in the original model require
more than 10 iterations to converge.
For this reason, ARIMA models cannot be identified by "backward stepwise" approach
that includes both AR and MA terms. In other words, you cannot begin by including
several terms of each kind and then throwing out the ones whose estimated
coefficients are not significant. Instead, you normally follow a "forward stepwise"
approach, adding terms of one kind or the other as indicated by the appearance of
the ACF and PACF plots.
Unit roots: If a series is grossly under- or overdifferenced--i.e., if a whole order of
differencing needs to be added or cancelled, this is often signalled by a "unit root" in
the estimated AR or MA coefficients of the model. An AR(1) model is said to have a
unit root if the estimated AR(1) coefficient is almost exactly equal to 1. (By "exactly
equal " I really mean not significantly different from, in terms of the coefficient's own
standard error.) When this happens, it means that the AR(1) term is precisely
mimicking a first difference, in which case you should remove the AR(1) term and
add an order of differencing instead. (This is exactly what would happen if you fitted
an AR(1) model to the undifferenced UNITS series, as noted earlier.) In a higherorder AR model, a unit root exists in the AR part of the model if the sum of the AR
coefficients is exactly equal to 1. In this case you should reduce the order of the AR
term by 1 and add an order of differencing. A time series with a unit root in the AR
coefficients is nonstationary--i.e., it needs a higher order of differencing.

Rule 9: If there is a unit root in the AR part of the model--i.e., if the
sum of the AR coefficients is almost exactly 1--you should reduce the
number of AR terms by one and increase the order of differencing by
one.
Similarly, an MA(1) model is said to have a unit root if the estimated MA(1)
coefficient is exactly equal to 1. When this happens, it means that the MA(1) term is
exactly cancelling a first difference, in which case, you should remove the MA(1)
term and also reduce the order of differencing by one. In a higher-order MA model, a
unit root exists if the sum of the MA coefficients is exactly equal to 1.

Rule 10: If there is a unit root in the MA part of the model--i.e., if the
sum of the MA coefficients is almost exactly 1--you should reduce the
number of MA terms by one and reduce the order of differencing by
one.
For example, if you fit a linear exponential smoothing model (an ARIMA(0,2,2)
model) when a simple exponential smoothing model (an ARIMA(0,1,1) model) would
have been sufficient, you may find that the sum of the two MA coefficients is very
nearly equal to 1. By reducing the MA order and the order of differencing by one
each, you obtain the more appropriate SES model. A forecasting model with a unit
root in the estimated MA coefficients is said to be noninvertible, meaning that the
residuals of the model cannot be considered as estimates of the "true" random noise
that generated the time series.
Another symptom of a unit root is that the forecasts of the model may "blow up" or
otherwise behave bizarrely. If the time series plot of the longer-term forecasts of the
model looks strange, you should check the estimated coefficients of your model for
the presence of a unit root.

Rule 11: If the long-term forecasts appear erratic or unstable, there
may be a unit root in the AR or MA coefficients.
None of these problems arose with the two models fitted here, because we were
careful to start with plausible orders of differencing and appropriate numbers of AR
and MA coefficients by studying the ACF and PACF models.
Go to next topic: Estimation of ARIMA models
Estimation of ARIMA models
Linear versus nonlinear least squares
Mean versus constant
Backforecasting
Linear versus nonlinear least squares
ARIMA models which include only AR terms are special cases of linear regression
models, hence they can be fitted by ordinary least squares.



AR forecasts are a linear function of the coefficients as well as a linear
function of past data.
In principle, least-squares estimates of AR coefficients can be exactly
calculated from autocorrelations in a single "iteration".
In practice, you can fit an AR model in the Multiple Regression procedure-just regress DIFF(Y) (or whatever) on lags of itself. (But you would get
slightly different results from the ARIMA procedure--see below!)
While you can use regression to estimate some ARIMA models- at
least the AR (p) component- you generally have to use the
regressions in several “iterations” as noted above to reestimate the
coefficients. This changes the meaning , accuracy, and usefulness of
the t-statisitcs and prob-values that we normally use in regressions.
ARIMA models which include MA terms are similar to regression models, but can't be
fitted by ordinary least squares:


Forecasts are a linear function of past data, but they are nonlinear functions
of coefficients--e.g., an ARIMA(0,1,1) model without constant is an
exponentially weighted moving average:
...in which the forecasts are a nonlinear function of the MA(1) parameter
("theta").
Another way to look at the problem: you can't fit MA models using ordinary
multiple regression because there's no way to specify ERRORS as an
independent variable--the errors are not known until the model is fitted! They



need to be calculated sequentially, period by period, given the current
parameter estimates.
MA models therefore require a nonlinear estimation algorithm to be used,
similar to the "Solver" algorithm in Excel.
The algorithm uses a search process that typically requires 5 to 10
"iterations" and occasionally may not converge.
You can adjust the tolerances for determining step sizes and stopping criteria
for search (although default values are usually OK).
"Mean" versus "constant
The "mean" and the "constant" in ARIMA model-fitting results are different numbers
whenever the model includes AR terms. Suppose that you fit an ARIMA model to Y in
which p is the number of autoregressive terms. (Assume for convenience that there
are no MA terms.) Let y denote the differenced (stationarized) version of Y--e.g.,
y(t) = Y(t)-Y(t-1) if one nonseasonal difference was used. Then the AR(p) forecasting
equation for y is:
This is just an ordinary multiple regression model in which "mu" is the constant term,
"phi-1" is the coefficient of the first lag of y, and so on.
Now, internally, the software converts this slope-intercept form of the regression
equation to an equivalent form in terms of deviations from the mean. Let m denote
the mean of the stationarized series y. Then the p-order autoregressive equation can
be written in terms of deviations from the mean as:
By collecting all the constant terms in this equation, we see it is equivalent to the
"mu" form of the equation if:
The software actually estimates "m" (along with the other model parameters) and
reports this as the MEAN in the model-fitting results, along with its standard error
and t-statistic, etc. The CONSTANT (i.e., "mu") is then calculated according to the
preceding formula, i.e.,
CONSTANT = MEAN*(1 - sum of AR coefficients)
If the model does not contain any AR terms, the MEAN and the CONSTANT are
identical.
In a model with one order of nonseasonal differencing (only), the MEAN is the trend
factor (average period-to-period change). In a model with one order of seasonal
differencing (only), the MEAN is the annual trend factor (average year-to-year
change).
"Backforecasting"










The basic problem: an ARIMA model (or other time series model) predicts
future values of the time series from past values--but how should the
forecasting equation be initialized to make a forecast for the very first
observation? (Actually, AR models can be initialized by dropping the first few
observations--although this is inefficient and wastes data-- but MA models
require an estimate of a prior error before they can make the first forecast.)
Strange but true: a stationary time series looks the same going forward or
backward in time, therefore...
The same model that predicts the future of a series can also be used to
predict its past.
The solution: to squeeze the most information out of the available data, the
best way to initialize an ARIMA model (or any time series forecasting model)
is to use backward forecasting ("backforecasting") to obtain estimates of data
values prior to period 1.
When you use the backforecasting option in ARIMA estimation, the search
algorithm actually makes two passes through the data on each iteration: first
a backward pass is made to estimate prior data values using the current
parameter estimates, then the estimated prior data values are used to
initialize the forecasting equation for a forward pass through the data.
If you DON'T use the backforecasting option, the forecasting equation is
initialized by assuming that prior values of the stationarized series were equal
to the mean.
If you DO use the backforecasting option, then the backforecasts that are
used to initialize the model are implicit parameters of the model, which must
be estimated along with the AR and MA coefficients. The number of additional
implicit parameters is roughly equal to the highest lag in the model--usually 2
or 3 for a nonseasonal model, and s+1 or 2s+1 for a seasonal model with
seasonality=s. (If the model includes both a seasonal difference and a
seasonal AR or MA term, it needs two season's worth of prior values to start
up!)
Note that with either backforecasting option, an AR model is estimated in a
different way than it would be estimated in the Multiple Regression procedure
(missing values are not merely ignored--they are replaced either with an
estimate of the mean or with backforecasts), hence an AR model fitted in the
ARIMA procedure will never yield exactly the same parameter estimates as an
AR model fitted in the Multiple Regression procedure.
Conventional wisdom: turn backforecasting OFF when you are unsure if the
current model is valid, turn it ON to get final parameter estimates once you're
reasonably sure the model is valid.
If the model is mis-specified, backforecasting may lead to failures of the
parameter estimates to converge and/or to unit-root problems.