Time Series Modelling with Auto Regression Model and Forecasting

The present report scrutinizes the relationship between the market return (S&P 500) and a company’s return. General Dynamics (GD) has been selected as the company in the research. Monthly market returns of GD and S&P 500 have been collected from Yahoo finance for a period of four years. The time period of returns is 01-04-2015 to 01-03-2019. Market return of S&P 500 has been described using descriptive any graphical methods. In second part of the report the dependence of GD’s return on previous returns and previous market returns has been assessed choosing significant number of lags. At last, an auto-regressive model has been constructed with monthly returns of GD stock.

Time Series Modelling with Auto Regression Model and Forecasting
1

Executive Summary
The present report scrutinizes the relationship between the market return (S&P 500) and a
company’s return. General Dynamics (GD) has been selected as the company in the research.
Monthly market returns of GD and S&P 500 have been collected from Yahoo finance for a
period of four years. The time period of returns is 01-04-2015 to 01-03-2019. Market return
of S&P 500 has been described using descriptive any graphical methods. In second part of the
report the dependence of GD’s return on previous returns and previous market returns has
been assessed choosing significant number of lags. At last, an auto-regressive model has been
constructed with monthly returns of GD stock.
2

Part A
a) Time series plots and histograms have been constructed for monthly returns of GD stock.
Square of monthly returns, and absolute market returns have also been presented in time
series and histogram plots (Montgomery, Jennings, and Kulahci, 2016).
Figure 1: Time Series plot of monthly return on GD stock from 01-05-2015 to 01-03-2019
Figure 2: Histogram for monthly return on GD stock from 01-05-2015 to 01-03-2019
3

Figure 3: Time Series plot of square of monthly return on GD stock from 01-05-2015 to 01-03-2019
Figure 4: Histogram for square of monthly return on GD stock from 01-05-2015 to 01-03-2019
4

Figure 5: Time Series plot of absolute monthly return on GD stock from 01-05-2015 to 01-03-2019
Figure 6: Histogram for absolute monthly return on GD stock from 01-05-2015 to 01-03-2019
b) Descriptive measures of central tendency have been presented below in Table 1.
Table 1: Descriptive Summary of monthly market returns of GD stock and S&P 500
Descriptive Statistics r(t) f(t) r(t)^2 f(t)^2 |r(t)| |f(t)|
Mean 0.0045 0.0065 0.0034 0.0012 0.0428 0.0254
Standard Error 0.0086 0.0050 0.0009 0.0003 0.0059 0.0035
Median 0.0020 0.0091 0.0008 0.0004 0.0289 0.0192
Standard Deviation 0.0590 0.0344 0.0064 0.0020 0.0403 0.0238
Sample Variance 0.0035 0.0012 0.0000 0.0000 0.0016 0.0006
Kurtosis 2.0205 1.5137 9.1494 5.9100 2.8949 1.0044
Skewness -0.6904 -0.6567 3.0469 2.4163 1.6786 1.2377
Range 0.3220 0.1760 0.0292 0.0093 0.1706 0.0959
Minimum -0.1708 -0.0963 0.0000 0.0000 0.0001 0.0004
Maximum 0.1512 0.0797 0.0292 0.0093 0.1708 0.0963
Sum 0.2092 0.3068 0.1609 0.0565 2.0125 1.1952
Count 47 47 47 47 47 47
First Quartile -0.0222 -0.0012 0.0003 0.0000 0.0162 0.0048
Third Quartile 0.0414 0.0217 0.0037 0.0012 0.0605 0.0344
Interquartile Range 0.0636 0.0229 0.0034 0.0012 0.0443 0.0296
Coefficient of Variation 7.55% 18.97% 53.14% 60.31%
106.20
%
106.75
%
NOTE: r (t) represents market return of GD stock, and f (t) represents the market returns of S&P 500
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Findings: Monthly market returns of GD stock (SD = 0.059) are more dispersed compared to
S&P 500 (SD = 0.034), for obvious reasons, as S&P 500 is the aggregate of all the stocks
traded in the stock exchange. Compared to the market, GD stock can be noted more volatile,
though, from Figure 1 it can be noted that volatility up to 01-01-2018 was more or less in line
with S&P 500. But, a sharp dip and downward trend can be clearly identified within the time
frame of 1-01-2018 to 01-03-2019. This particular observation can clearly be identified from
Figure 2 and Figure 3 presenting the square and absolute returns. Both GD and S&P500 are
found to be positively skewed. Due to recent fluctuations in returns of GD, skewness for GD
stock is noted to be slightly greater than S&P500. An average monthly return of GD is well
below the average S&P500 return, indicating that the stock performs below par the entire
market. Considering the median of returns, it is noted that median return for GD is way below
S&P500 (Addo, and Sunzuoye, 2013, pp.15-21). This particular statistical observation
indicated that average of GD stock is affected by few outlier or unusual higher returns.
Part B
Using multiple linear regression analysis, the mathematical model yt =c +∑
i=1
k
yt −i +∑
i=0
j
f t −i+ut
has been estimated with appropriate number of lags by misspecification tests.
Market returns from 01-05-2015 to 01-03-2019 have been considered for this regression
analysis.
First, the autocorrelations for market return for GD (r (t)) and S&P500 (f (t)) stocks have
been evaluated for 11 lags (number of observations = 47). No significant autocorrelation
(|ρ|≥3 ) was found for f (t). Significant autocorrelations were noted for lag 1, 2, and 6 for r (t).
The scholar included 6 lags for r (t) and zero lag for f (t) variables in the first ARDL
regression model (Belloumi, 2014, pp.269-287).
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Model: The ARDL model was framed as
Y t =c +∑
i =1
k
βi Y t −i+∑
i=0
j
ηi f t−i+ut
where ut
represents residuals of the model. Here, Y (t) is considered in place of r (t). k = number of
lags for Y (t), and j = number of lags for f (t).
Let ut (t =1, 2, 3… T) be the estimated residuals from the regression model, where,
ut=ρ1 ut−1+ ρ2 ut−2+ ρ3 ut −3+. .. .+ ρk ut−k +et Where ρi ' s denote the auto correlations at
ith
order.
Findings: first, the auto correlations were evaluated for Y (t) and f (t), which have been
presented in Table 2. From Table 2 it is noted that f (t) has no serial correlation, whereas Y (t)
has significant auto correlations for lag 1, 2, and 6.
Table 2: Auto Correlation for 11 lags for f (t) and Y (t)
Autocorrelations for f(t) Autocorrelations for Y(t)
Lag Autocorr St.Err
La
g Autocorr St.Err
1 -0.1260 0.1459 1 -0.4132 0.1459
2 -0.0748 0.1459 2 0.3494 0.1459
3 -0.2148 0.1459 3 -0.2114 0.1459
4 -0.0567 0.1459 4 0.2590 0.1459
5 0.0447 0.1459 5 -0.2134 0.1459
6 0.0121 0.1459 6 0.3409 0.1459
7 0.0218 0.1459 7 -0.1267 0.1459
8 0.1616 0.1459 8 0.0436 0.1459
9 0.0240 0.1459 9 -0.0164 0.1459
10 0.0272 0.1459 10 0.0775 0.1459
11 -0.1595 0.1459 11 -0.2361 0.1459
First empirical regression model:
This model consisted of independent variables Y(t)_Lag1, Y(t)_Lag2, f(t). We tested the
model for misspecifications and decided the proper number of lags with test for no-
autocorrelation (Breusch-Godfrey Test).
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Table 3: Multiple Regression Model with p = 2, q = 0 lags for Y (t) and f (t)
Summary measures
Multiple R 0.7125
R-Square 0.5076
Adj R-Square 0.4716
StErr of Est 0.0438
ANOVA Table
Source df SS MS F p-value
Explained 3 0.0811 0.0270 14.0911 0.0000
Unexplained 41 0.0786 0.0019
Regression coefficients
Coeffi cient Std Err t-value p-value Lower limit Upper limit
Constant -0.0034 0.0068 -0.5000 0.6198 -0.0171 0.0103
Y(t)_Lag1 -0.1482 0.1256 -1.1800 0.2448 -0.4019 0.1054
Y(t)_Lag2 0.1984 0.1204 1.6477 0.1071 -0.0448 0.4416
f(t) 0.9930 0.1996 4.9759 0.0000 0.5900 1.3960
Hypothesis Testing: Autocorrelation (Breusch-Godfrey Test):
For investigating serial correlation, an auxiliary regression model was framed as
ut=ρ1 ut −1+ ρ2 ut −2+ ρ3 ut −3+ ρ4 ut −4 + ρ5 ut −5+ ρ6 ut−6+ δ1 Y (t ) _Lag1 + δ2 Y (t )_Lag2+ η1 f (t )+ et
The null hypothesis assuming no auto correlation was framed as below,
H0 : ρ1=ρ2=. ..=ρ6=0 was tested against the alternate hypothesis H A : ρi≠0 for at least
one value of i=1,2 ,…6.
Test statistics of Breusch-Godfrey Test is LM = ( n− p ) R2 ~ χ 2 ( p ) , where n denoted the
number of observation in the regression model, p is the number of lags of residuals, and R2
is the coefficient of determination in the auxiliary regression model.
LM-STAT = 7.28, DF = 6, P-Value = 0.295 > 0.05
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Hence, the null hypothesis failed to get rejected at 5% level of significance. So, no
autocorrelation was noted in the model.
But, from Table 3 it can be noted that lag 1 of Y (t) was not a significant predictor of Y (t).
Therefore, the model was not statistically adequate. The model was reconstructed with
Y(t)_lag 2 and f (t) as the two predictors.
Table 4: Multiple Regression Model with p = 1, q = 0 lags for Y (t) and f (t)
Summary measures
Multiple R 0.7007
R-Square 0.4909
Adj R-Square 0.4667
StErr of Est 0.0440
ANOVA Table
Source df SS MS F p-value
Explained 2 0.0784 0.0392 20.2512 0.0000
Unexplained 42 0.0813 0.0019
Regression coefficients
Coeffi cient Std Err t-value p-value Lower limit Upper limit
Constant -0.0048 0.0067 -0.7116 0.4807 -0.0183 0.0088
Y(t)_Lag2 0.2538 0.1114 2.2775 0.0279 0.0289 0.4787
f(t) 1.0598 0.1922 5.5131 0.0000 0.6719 1.4478
The predictors were found to be statistically significant in the model.
Hypothesis Testing: Autocorrelation (Breusch-Godfrey Test):
For investigating serial correlation, an auxiliary regression model was framed as
ut=ρ1 ut−1+ ρ2 ut−2+ρ3 ut−3+ ρ4 ut −4 + ρ5 ut−5+ ρ6 ut−6+ δ1 Y (t ) _Lag1 + η1 f (t )+ et
The null hypothesis assuming no auto correlation was framed as below,
H0 : ρ1=ρ2=. ..=ρ6=0 was tested against the alternate hypothesis H A : ρi≠0 for at least
one value of i=1,2 ,…6.
9

Test statistics of Breusch-Godfrey Test is LM = ( n− p ) R2 ~ χ 2 ( p ) , where n denoted the
number of observation in the regression model, p is the number of lags of residuals, and R2
is the coefficient of determination in the auxiliary regression model.
LM-STAT = 8.75, DF = 6, P-Value = 0.188 > 0.05
Hence, the null hypothesis failed to get rejected at 5% level of significance. So, no
autocorrelation was noted in the model. The final model is
Y ( t )=0 . 2538∗Y ( t )Lag 2+1. 0598∗f (t )−0 . 0048 .
Part C
Using Linear Regression Analysis an AR (p) Model was estimated choosing the appropriate
number of lags. Misspecification testing was done on the model to choose number of
appropriate lags. Also, a statistically significant regression model has been constructed.
Using Table 1 for autocorrelation of Y(t), the first model has been constructed with lag 1
(A(1)). The detailed model is as below.
Table 5: Autoregressive model with one lagged variable
Summary measures
Multiple R 0.4137
R-Square 0.1711
Adj R-Square 0.1523
StErr of Est 0.0549
ANOVA Table
Source df SS MS F p-value
Explained 1 0.0273 0.0273 9.0841 0.0043
Unexplained 44 0.1324 0.0030
Regression coefficients
Coeffi cient Std Err t-value p-value Lower limit Upper limit
Constant 0.0060 0.0081 0.7437 0.4610 -0.0103 0.0224
Y(t)_Lag1 -0.4135 0.1372 -3.0140 0.0043 -0.6899 -0.1370
10

Hypothesis Testing: Autocorrelation (Breusch-Godfrey Test):
For investigating serial correlation, an auxiliary regression model was framed as
ut=ρ1 ut−1+ ρ2 ut−2+ρ3 ut −3+ρ4 ut −4 + ρ5 ut−5+ ρ6 ut−6+ δ1 Y (t ) _Lag1 + et
The null hypothesis assuming no auto correlation was framed as below,
H0 : ρ1=ρ2=. ..=ρ6=0 was tested against the alternate hypothesis H A : ρi≠0 for at least
one value of i=1,2 ,…6.
Test statistics of Breusch-Godfrey Test is LM = ( n− p ) R2 ~ χ 2 ( p ) , where n denoted the
number of observation in the regression model, p is the number of lags of residuals, and R2
is the coefficient of determination in the auxiliary regression model (Hyun et al., 2010,
pp.399-404). LM-STAT = 15.33, DF = 6, P-Value = 0.018 < 0.05
Hence, the null hypothesis was rejected at 5% level of significance. So, autocorrelation was
noted in the model. More lags have to be added to the model.
The second model has been constructed with lag 1 and lag 2 (A(2)). The detailed model is as
below.
Table 6: Autoregressive model with two lagged variables
Summary measures
Multiple R 0.4586
R-Square 0.2103
Adj R-Square 0.1727
StErr of Est 0.0548
ANOVA Table
Source df SS MS F p-value
Explained 2 0.0336 0.0168 5.5928 0.0070
Unexplained 42 0.1261 0.0030
Regression coefficients
Coeffi cient Std Err t-value p-value Lower limit Upper limit
Constant 0.0043 0.0083 0.5220 0.6044 -0.0123 0.0210
Y(t)_Lag1 -0.3256 0.1507 -2.1609 0.0365 -0.6298 -0.0215
Y(t)_Lag2 0.2150 0.1506 1.4272 0.1609 -0.0890 0.5190
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Breusch-Godfrey Test with auxiliary regression model was tested at 5% level of
significance.
ut=ρ1 ut −1+ ρ2 ut−2+ ρ3 ut −3+ ρ4 ut−4 + ρ5 ut−5+ ρ6 ut−6+ δ1 Y (t ) _Lag1 + δ2 Y (t )_Lag2+et
LM-STAT = 6.92, DF = 6, P-Value = 0.3281 > 0.05
Hence, the null hypothesis failed to get rejected at 5% level of significance. So, no
autocorrelation was noted in the model.
Y(t)_Lag 2 was found to be statistically insignificant in the model. Excluding Y(t)_Lag 2
from the model, we included Y(t)_Lag 6. The new regression model was constructed as
below.
Table 7: Autoregressive model with two lagged variables, Y(t)_Lag2 and Y(t)_Lag 6
Summary measures
Multiple R 0.5538
R-Square 0.3067
Adj R-Square 0.2702
StErr of Est 0.0521
ANOVA Table
Source df SS MS F p-value
Explained 2 0.0456 0.0228 8.4062 0.0009
Unexplained 38 0.1030 0.0027
Regression coefficients
Coeffi cient Std Err t-value p-value Lower limit Upper limit
Constant -0.0001 0.0084 -0.0145 0.9885 -0.0171 0.0169
Y(t)_Lag1 -0.2940 0.1397 -2.1045 0.0420 -0.5768 -0.0112
Y(t)_Lag6 0.4939 0.1828 2.7012 0.0103 0.1238 0.8640
Breusch-Godfrey Test with auxiliary regression model was tested at 5% level of
significance.
ut=ρ1 ut −1+ ρ2 ut−2+ ρ3 ut−3+ ρ4 ut−4 + ρ5 ut−5+ ρ6 ut−6+ δ1 Y ( t ) _Lag1 + δ2 Y (t ) _Lag6+et
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LM-STAT = 9.55, DF = 6, P-Value = 0.1488 > 0.05
Hence, the null hypothesis failed to get rejected at 5% level of significance. So, no
autocorrelation was noted in the model and the model is also statistically significant.
No-Heteroscedasticity: Breusch Pagan Test:
For investigating Heteroscedasticity, an auxiliary regression model was framed as
ut
2=δ0+ δ1 Y (t )_Lag1 + δ2 Y (t ) _Lag6+et
The null hypothesis assuming no Heteroscedasticity was framed as below,
H0 :δ 1=δ2=0 was tested against the alternate hypothesis H A : δi≠0 for at least one value of
i=1,2 .
Test statistics of Breusch-Pagan Test is LM = ( n ) R2 ~ χ2 ( k ) , where n denoted the number of
observation in the regression model, and R2
is the coefficient of determination in the
auxiliary regression model (Baltagi, Jung, and Song, 2010, pp.122-124).
LM-STAT = 4.99, DF = 4, P-Value (Chi-square with 4 df) = 0.287 > 0.05
Hence, the null hypothesis failed to get rejected at 5% level of significance. So, no
Heteroscedasticity was noted in the model.
Normality of Residuals: Jarque Berra Test:
Skewness = -0.0186, Excess Kurtosis = 0.604, N = 41,
Null hypothesis: Residuals are normally distributed
Alternate hypothesis: Residuals are not normally distributed
13

JB-STAT =
N
6 ( S2+ K 2
4 ) =
0.625
P-VALUE = Chi-Square (2) = 0.731 > 0.05.
Hence, the null hypothesis failed to get rejected, implying that the residuals are normally
distributed.
Figure 7: Residual versus Y(t) graph
14

Figure 8: Residual versus Fitted Values
The final auto regression model is Y ( t )=0 . 4939∗Y ( t )Lag 6−0 . 2940∗Y (t )Lag 1−0 . 0001
where Y(t) represents monthly returns for GD stock.
Figure 9: 5-Year Monthly basis forecasting from Auto regressive model
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Table 8: 5-Year forecasted Monthly Data
Date Y(t) Date Y(t)
01-04-2019 -0.0828 01-10-2021 -0.0085
01-05-2019 0.0583 01-11-2021 0.0083
01-06-2019 -0.0974 01-12-2021 -0.0102
01-07-2019 0.0705 01-01-2022 0.0112
01-08-2019 -0.0236 01-02-2022 -0.0107
01-09-2019 0.0041 01-03-2022 0.0079
01-10-2019 -0.0422 01-04-2022 -0.0067
01-11-2019 0.0411 01-05-2022 0.0059
01-12-2019 -0.0603 01-06-2022 -0.0069
01-01-2020 0.0524 01-07-2022 0.0075
01-02-2020 -0.0272 01-08-2022 -0.0076
01-03-2020 0.0099 01-09-2022 0.0060
01-04-2020 -0.0239 01-10-2022 -0.0052
01-05-2020 0.0272 01-11-2022 0.0043
01-06-2020 -0.0379 01-12-2022 -0.0048
01-07-2020 0.0369 01-01-2023 0.0050
01-08-2020 -0.0244 01-02-2023 -0.0053
01-09-2020 0.0119 01-03-2023 0.0044
01-10-2020 -0.0154 01-04-2023 -0.0040
01-11-2020 0.0178 01-05-2023 0.0032
01-12-2020 -0.0241 01-06-2023 -0.0034
01-01-2021 0.0252 01-07-2023 0.0033
01-02-2021 -0.0196 01-08-2023 -0.0037
01-03-2021 0.0115 01-09-2023 0.0032
01-04-2021 -0.0111 01-10-2023 -0.0030
01-05-2021 0.0120 01-11-2023 0.0023
01-06-2021 -0.0155 01-12-2023 -0.0025
01-07-2021 0.0169 01-01-2024 0.0023
01-08-2021 -0.0148 01-02-2024 -0.0026
01-09-2021 0.0099 01-03-2024 0.0022
Interpretations:
The five year monthly forecasted data has been provided in Table 8. Forecasted values can be
evaluated as follows.
On 01-04-2019,
16

Y (t )=0 . 4939∗Y ( t )Lag 6−0 . 2940∗Y (t )Lag 1−0 . 0001
=0 . 4939∗Y (01-10-2018)−0. 2940∗(01-03-2019)-0 . 0001
=0 . 4939*(-0 . 1708 )-0. 2940*(-0 . 0055 )-0 . 0001
=−0. 0828
=−8. 28 %
With increase in time period, forecasting error increases. After an optimal length of time the
previous periods become less informative with the current regression model
Y (t )=0 . 4939∗Y ( t )Lag 6−0 . 2940∗Y ( t )Lag 1−0 . 0001 .
Forecast variance converges to the unconditional variance with increase in time. This
particular trend can also be noted in Figure 9, where after 01-06-2020 the trend smoothens to
a flat curve close to the zero return level. Weights decline exponentially in the forecasting
trend, and ideally we should use most recent observations as they weighted most.
Part D
The problems of regression and autoregressive model selection are closely related. Indeed,
many of the proposed solutions can be applied equally well to both problems. One of the
leading selection methods is the Akaike information criterion to choose proper number of
lags. Instead of this criterion, we have used misspecification test (Breusch-Godfrey Test) to
check the auxiliary regression model for serial correlation (Nkoro, and Uko, 2016, pp.63-91).
In general, it's not easy to determine a model that best represents the process of data
generation, and it's not uncommon to estimate more models at early stage. The final selection
model is based on a set of diagnostic test criteria for the best model. These criteria include the
(1) coefficient t-test, (2) the Residue Analysis (3) model selection criteria.
17

The General Dynamic stock (GD) significantly depends on S&P 500 without any time lag
and on the previous market returns of GD for one period of lag. Significant estimation auto
regressive model with distributed lag evaluated a model significantly depending on every
prior 2nd month and on the present S&P500 value. The auto regressive model with exogenous
character was found to dependent on previous month and on every prior 6th month. The
primary disadvantage of forecasting with this model was ever increasing forecasting error
with increase in time.
References
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