2. Forecasting Project ii
Table of Contents
Overview of the Data....................................................................................................... 1
Autoregressive Estimation................................................................................................ 3
AR(12) ............................................................................................................................................... 3
AR(12) with Time Trend..................................................................................................................... 4
ARMA ............................................................................................................................................... 5
Volatility Adjustment....................................................................................................... 6
Accelerating Growth........................................................................................................................... 6
Month-to-month Changes ................................................................................................................... 7
Forecast for 2007 ............................................................................................................. 8
3. Yiqun Li
Macroeconometrics
Ai Deng
April 2015
Forecasting Project 1
Forecasting Project
Homework 4
This paper briefly discusses how I used macroeconometric techniques to forecast the data of
all 12 months 2007, given a time series data ranging from Jan 1996 to Dec 2006. Before the
projection, I had no knowledge of what was the source of the data.
The performance of the forecast is shown in the appendix. The paper is composed by three
sections, beginning with an overview of the data, which is followed by autoregressive
estimation, and finally a volatility/variance adjustment. I used Stata 13 to complete all the
graphing and computation.
Overview of the Data
Before applying any econometrical or statistical techniques, it is necessary to take an
overview of the data, which serves as a shortcut to determine what forecasting models will
100000200000300000400000
Y
1992m1 1994m1 1996m1 1998m1 2000m1 2002m1 2004m1 2006m1
month
Graph 1
4. Forecasting Project 2
be more efficient and unbiased.
Graph 1 plots the data from 1996 to 2006. From this graph, it is easy to draw two
preliminary conclusions:
1. There is a seasonal pattern with increasing trend.
2. The volatility/variance is increasing.
We observe from the graph that each season consists of 12 periods. To confirm this
preliminary conclusion, simply find the AC and PAC of the variable.
-1 0 1 -1 0 1
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-------------------------------------------------------------------------------
1 0.8582 0.8945 134.8 0.0000 |------ |-------
2 0.8196 0.3711 258.42 0.0000 |------ |--
3 0.8267 0.4129 384.92 0.0000 |------ |---
4 0.8251 0.3453 511.64 0.0000 |------ |--
5 0.8176 0.3730 636.78 0.0000 |------ |--
6 0.7884 0.2241 753.79 0.0000 |------ |-
7 0.7844 0.3617 870.3 0.0000 |------ |--
8 0.7588 0.2498 979.98 0.0000 |------ |-
9 0.7295 0.1502 1081.9 0.0000 |----- |-
10 0.6871 -0.2350 1172.9 0.0000 |----- -|
11 0.6986 -0.1914 1267.5 0.0000 |----- -|
12 0.8003 0.9617 1392.4 0.0000 |------ |-------
13 0.6681 -0.4277 1480 0.0000 |----- ---|
14 0.6311 -0.2593 1558.6 0.0000 |----- --|
15 0.6330 -0.3428 1638.1 0.0000 |----- --|
From
the
PAC,
two
spikes
are
outstanding
at
Lag
1
and
Lag
12.
However,
this
is
not
sufficient
to
determine
which
periods
actually
matter,
and
is
further
confirmed
by
using
detrended
data.
-1 0 1 -1 0 1
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-------------------------------------------------------------------------------
1 -0.0516 -0.0547 .48784 0.4849 | |
2 -0.2456 -0.2647 11.585 0.0031 -| --|
3 -0.0509 -0.0946 12.065 0.0072 | |
4 0.0558 -0.0218 12.645 0.0131 | |
5 0.1469 0.1328 16.684 0.0051 |- |-
6 0.0239 0.0587 16.791 0.0101 | |
7 0.1405 0.2605 20.528 0.0045 |- |--
8 0.0732 0.1860 21.549 0.0058 | |-
9 -0.0501 0.1019 22.03 0.0088 | |
10 -0.2864 -0.2995 37.841 0.0000 --| --|
11 -0.0934 -0.2863 39.531 0.0000 | --|
12 0.8758 0.9565 189.09 0.0000 |------- |-------
13 -0.0768 -0.4584 190.25 0.0000 | ---|
14 -0.2424 -0.2845 201.85 0.0000 -| --|
15 -0.0929 -0.3751 203.56 0.0000 | ---|
5. Forecasting Project 3
Once again, PAC graph shows significant autocorrelation at Lag 12.
Therefore, two preliminary models can be constructed:
1. AR(12)
2. MA(12) with a time trend
3. ARMA
The AR(12) model is simply a 12 periods lagged auto-regression. This model is based on the
assumption that the time series is a result of the sum of seasonal increments and the
increment is stationary namely; the process is difference-stationary. The ARMA model is
based on the same assumption, whereas the MA(12) model assumes that the process is
trend-stationary.
Autoregressive Estimation
Having constructed the preliminary models, this part estimates the autoregression and tests
the stationarity assumption of each model to compare the predictive efficiency of the
models. Lastly, this part applies pseudo out-of-sample method.
AR(12)
Since the given data starts at a level that is way above 0, it is reasonable to include a
constant term in the model. The model is therefore:
𝑌 = 7225.12 + 1.022𝑌!" + 𝜀
Running the pseudo out-of-sample method, we can compare the prediction with actual data
(Graph 2).
6. Forecasting Project 4
Noticeably, this model overall fits the actual well; however, it performs unsatisfactorily in
predicting the month-to-month change in a year, which is also the most difficult part to
predict. Though we may be able to tell whether there will be a climb or a fall from one
month to another, it is hard to tell the extent. Conceivably, this issue will also occur in the
MR(12) model. To address the issue, it is necessary to analyze the volatility of the data,
which will be discussed later.
To test whether the process is difference stationary, simply use Dickey-Fuller test on the first
difference of the original data.
Dickey-Fuller test for unit root Number of obs = 178
---------- Interpolated Dickey-Fuller ---------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -20.034 -3.484 -2.885 -2.575
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0000
According to the test results, the process is difference stationary; therefore, the AR(12)
model is reasonable.
AR(12) with Time Trend
This model is basically an MA model that assumes the data are just a time series that
fluctuate along a time trend.
𝑌 = −243136.3 + 1003.476𝑡 + 𝛾
𝛾 = .982016𝛾!" + 𝜀
Again, using the pseudo-out-of-sample method (Graph 3):
7. Forecasting Project 5
We can see that compared to AR(12) model, the MR(12) with time trend is more deviant
from the real data. The MA(12) term in the model has a coefficient of 0.98, which is less
than 1. This is contrary to our observation that the deviation from the time trend becomes
larger across time. Besides, the moving-average term does not estimate efficiently as it treats
all observations equally. It is reasonable to assume that when the process comes to a peak,
the deviation from its linear estimation is greater. This model is therefore abandoned.
ARMA
This autoregressive-moving-average model consists of a lag 12 autoregressive term and a lag
12 moving-average term. This model basically substitutes the time trend with an
autoregressive term.
𝑌 = 237695.6 + 0.9797656𝑌!" + 𝛾
𝛾 = 0.6399572𝛾!" + 𝜀
The original data is non-stationary whereas in this estimation, the coefficient of the
autoregressive term is smaller than 1. Even though it is very close to 1, we see it slow down
the growth of the pseudo-out-of-sample forecast (Graph 4):
The gap between the forecast and the real data become lager, this indicates that the
estimation is biased. The model is therefore biased.
After comparing all the alternatives, AR(12) model appears to be the most efficient model,
based on which we make adjustment for volatility.
8. Forecasting Project 6
Volatility Adjustment
The following table shows that the standard deviation of each year has increased since the
beginning of the given time series.
1992|15294.081
1993|17866.066
1994|19212.123
1995|18420.228
1996|17466.754
1997|17656.24
1998|20267.972
1999|23834.002
2000|19410.558
2001|20375.55
2002|21268.23
2003|23225.521
2004|25354.954
2005|27391.494
2006|25503.455
Two behaviors of the process can explain such phenomenon:
• Accelerating growth.
• Enlarging month-to-month changes.
Accelerating Growth
To test if there is an accelerating growth, simply introduce a squared autoregressive term
into the original AR(12) model. This yields the following estimation:
Source | SS df MS Number of obs = 168
-------------+------------------------------ F( 2, 165) = 5944.92
Model | 4.5624e+11 2 2.2812e+11 Prob > F = 0.0000
Residual | 6.3314e+09 165 38371828.8 R-squared = 0.9863
-------------+------------------------------ Adj R-squared = 0.9861
Total | 4.6257e+11 167 2.7699e+09 Root MSE = 6194.5
------------------------------------------------------------------------------
Y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Y |
L12. | .9496485 .0748648 12.68 0.000 .8018322 1.097465
|
Y_squared |
L12. | 1.58e-07 1.61e-07 0.98 0.329 -1.60e-07 4.75e-07
|
_cons | 15211.37 8435.846 1.80 0.073 -1444.753 31867.49
------------------------------------------------------------------------------
The coefficient for the squared autoregressive term is very close to zero and insignificant. It
is therefore reasonable to say that the growth is steady, and the increasing volatility is not
due to an accelerating growth.
9. Forecasting Project 7
Month-to-month Changes
To test if the month-to-month changes are truly enlarging, simply run a stationarity test on
the absolute month-to-month changes. If it is not stationary, then month-to-month changes
can hardly explain the increasing volatility.
To perform the test, take the absolute values of month-to-month changes first and then use
Augmented Dickey-Fuller test.
DF-GLS mu 1% Critical 5% Critical 10% Critical
[lags] Test Statistic Value Value Value
------------------------------------------------------------------------------
13 1.189 -2.589 -1.964 -1.657
12 1.150 -2.589 -1.972 -1.664
11 1.659 -2.589 -1.980 -1.672
10 -1.611 -2.589 -1.987 -1.679
9 -2.189 -2.589 -1.995 -1.686
8 -2.527 -2.589 -2.002 -1.693
7 -3.041 -2.589 -2.010 -1.700
6 -3.849 -2.589 -2.017 -1.707
5 -4.169 -2.589 -2.023 -1.713
4 -4.441 -2.589 -2.030 -1.719
3 -4.400 -2.589 -2.036 -1.724
2 -4.945 -2.589 -2.042 -1.729
1 -6.104 -2.589 -2.047 -1.734
Opt Lag (Ng-Perron seq t) = 12 with RMSE 5625.384
Min SC = 17.67238 at lag 12 with RMSE 5625.384
Min MAIC = 17.48427 at lag 12 with RMSE 5625.384
The test result proves that the month-to-month changes are not stationary.
Though we cannot firmly state that all the month-to-month changes in 2007 will be greater,
we are confident to say they will increase. Therefore, we add a lag 12 first difference term in
the AR(12) in order to adjust for the volatility. The regression gives us the final model:
𝑌 = 7296.199 + 1.022𝑌!" − 0.0047112𝑑𝑌!" + 𝜀
10. Forecasting Project 8
Forecast for 2007
According to this estimation, the forecast for 2007 is:
1/1/07 300036.6655
2/1/07 296234.9207
3/1/07 340786.5228
4/1/07 330978.1636
5/1/07 352252.8623
6/1/07 345596.4989
7/1/07 340576.7292
8/1/07 354092.1206
9/1/07 325187.4534
10/1/07 327431.0226
11/1/07 337816.8233
12/1/07 396156.7522
End.
