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ABERDEEN 2040
BU5565
Empirical Methods in Finance Research
Workshop 1 (week 28)
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1 . Define the concept of non-stationary. Discuss the different types of non-stationary data.
A Non-stationary time series does not have a constant mean and variance. In the lectures we
distinguish between a random walk and a random walk with a drift (a constant term). Overall, we
always need to transform data to obtain stationarity. There are different types of stationary:
• Strict Stationarity: Denotes that the joint distribution of any moments of any degree (i.e. expected
values, variances) within the process is never dependent on time.
• First-order stationarity: Indicates that the series have means that never change with time. Any
other moment like variance can change.
• Trend-stationary models fluctuate around a deterministic trend (the series mean). These
deterministic trends can be linear or quadratic, but the amplitude (height of one oscillation) of the
fluctuations neither increases nor decreases across the series.
• Difference-stationary models are models that need one or more differences to become
stationary. For example, differencing financial data like stock market data.
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1 . Define the concept of non-stationary. Discuss the different types of non-stationary data.
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2. In describing time series, we can use words such as “trend” and “seasonal”. Describe these
concepts.
Trend: long-term increase or decrease in the data. It does not have to be linear.
Some authors mention that a trend can also be called as “changing direction” when it might go from
an increasing trend to a decreasing trend:
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2. In describing time series, we can use words such as “trend” and “seasonal”. Describe these
concepts.
seasonal pattern: condition when a time series is affected by seasonal factors such as the time of
the year or the day of the week.
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3. Define differenced variable and its purpose.
• The dth differencing operator applied to a time series x is to create a new series z whose value at
time t is the difference between x(t + d) and x(t).
• This method works very well in removing trends and cycles.
• For example, first differencing applied to a series with a linear trend eliminates the trend while if
cycles of length d exist in a series, a dth difference will remove them.
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4. Define the concept of Autoregressive Process (AR). If I decide to model profits, as an AR (4)
process, how would I write the underlying function?
An AR process is when a value from a time series is regressed on previous values from that same
time series:
= 0 + 1−1 + (1)
The order of an autoregression is the number of immediately preceding values in the series that are
used to predict the value at the present time.
In this case, the above model is an AR with what order?
Answer: first -order autoregression written as AR(1).
So, with the same logic, a profit model with an AR(4) process would be:
(2)
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5. Discuss how you might use the Autocorrelation Function (ACF) and the Partial
Autocorrelation Function (PACF) to tell the type of underlying process.
• By looking at the ACF and the PACF plots of a differentiated series we can try to identify the AR
and/or MA terms that are needed.
The main issue?
Determining the order of an AR/Ma process from its autocorrelation function is difficult.
To resolve this issue, the Partial Autocorrelation Function (PACF) is introduced.
The ACF plot : bar chat of the coefficients of correlation between a time series and lags of itself.
The PACF: partial correlation coefficients between the series and lags of itself.
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6. A research plot different univariate time series Autocorrelation function and Partial
Autocorrelation function.
6.1) Define the below process
ACF : Coefficients decrease with the lag: AR process
PACF: Lag 1 is the only statistically significant coefficient.
Hence we have an AR(1) process.
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6. A research plot different univariate time series Autocorrelation function and Partial
Autocorrelation function.
6.2) Define the below process
ACF : A single coefficient different from zero is observed
PACF: A geometric decay is observed.
Hence we have an MA(1) process.
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7. Describe the steps behind the construction of an ARMA model using the Box-Jenkins
Methodology.
Box and Jenkins (1976) were the first to consider ARMA (and thereafter ARIMA) modelling in this
logical and coherent fashion. Their methodology consists of 3 steps:
Identification: We need to determine the specific order of the model (by using graphical
procedures, i.e. ACF and PACF)
Estimation: Estimation of the parameters of the model of size given in the first stage. This can be
done using the OLS or the maximum likelihood, depending on the model that we are testing.
Diagnostic checking: The fitted model is checked for inadequacies by checking for the
autocorrelation of the residual series and overfitting the model.
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8. Define the following univariate time-series models:
8.1: = −. − +
8.2: = −. − − . − +
8.3: = . − +
8.4: = . − + . − +
8.1) = −. − +
• MA(1)
8.2) = −. − − . − +
• MA(2)
8.3) = . − +
• AR(1)
8.4) = . − + . − +
• ARMA(1,1)
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9. Open the E-views file “GNPDEF.WF1”. The file contains information on quarterly
observations on U.S. GNP and the implicit price deflator for GNP for 1947 through 2019.
9.1) Plot the graph of the U.S. GNP series. What can you conclude?
• The series does not seem to have a constant mean and variance
• it seems that we are in the presence of a non-stationary series.
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9.2) In E-views run the Augmented Dickey-Fuller Test (ADF) for the above series. What can you
conclude?
Note: ADF definition
• ADF test is fundamentally a statistical significance test.
• Hence we have a hypothesis testing involved with a null and alternate hypothesis and as a result a
test statistic is computed and p-values get reported.
• It is from the test statistic and the p-value, you can make an inference as to whether a given series
is stationary or not. So, how exactly does the ADF test work?
➢ The ADF test belongs to a category of tests called ‘Unit Root Test’, which is the proper
method for testing the stationarity of a time series.
➢ So what does a ‘Unit Root’ mean? Unit root is a characteristic of a time series that
makes it non-stationary. Technically speaking, a unit root is said to exist in a time series
of the value of alpha = 1 in the below equation.
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9.2) In E-views run the Augmented Dickey-Fuller Test (ADF) for the above series. What can you
conclude?
Note: ADF definition
➢ So what does a ‘Unit Root’ mean? Unit root is a characteristic of a time series that
makes it non-stationary. Technically speaking, a unit root is said to exist in a time series
of the value of alpha = 1 in the below equation.
➢ where, Yt is the value of the time series at time ‘t’ and Xe is an exogenous variable (a
separate explanatory variable, which is also a time series). What does this mean to
us? The presence of a unit root means the time series is non-stationary. Besides, the
number of unit roots contained in the series corresponds to the number of differencing
operations required to make the series stationary.
➢ Hypotheses:
H0: A unit root is present in the model (it implies that the series is non-stationary)
Ha: The series is stationary.
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9.2) In E-views run the Augmented Dickey-Fuller Test (ADF) for the above series. What can you
conclude?
• ADF p-value=0.990 (or the t-stat of 1.4 falls in the non rejection area). We cannot reject the null hypothesis,
suggesting the presence of stationary.
• The second part of the output: the t-value of the lagged of the GNP series is 1.40062 with a p-value of 0.1624.
This suggests that the coefficient is not different from zero, and thus, supporting the hypothesis that the U.S.
GNP indicator is a random walk, or in other words, the series is non-stationary.
Null Hypothesis: GNPDEF has a unit root
Exogenous: Constant
Lag Length: 3 (Automatic - based on SIC, maxlag=15)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic 1.400662 0.9990
Test critical values: 1% level -3.452911
5% level -2.871367
10% level -2.572078
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(GNPDEF)
Method: Least Squares
Date: 01/11/21 Time: 13:14
Sample (adjusted): 1948Q1 2019Q4
Included observations: 288 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
GNPDEF(-1) 0.000395 0.000282 1.400662 0.1624
D(GNPDEF(-1)) 0.508514 0.058486 8.694607 0.0000
D(GNPDEF(-2)) 0.173378 0.065249 2.657187 0.0083
D(GNPDEF(-3)) 0.181805 0.059114 3.075523 0.0023
C 0.027744 0.016600 1.671256 0.0958
R-squared 0.698956 Mean dependent var 0.348257
Adjusted R-squared 0.694701 S.D. dependent var 0.247114
S.E. of regression 0.136540 Akaike info criterion -1.127191
Sum squared resid 5.276012 Schwarz criterion -1.063598
Log likelihood 167.3154 Hannan-Quinn criter. -1.101706
F-statistic 164.2652 Durbin-Watson stat 2.033520
Prob(F-statistic) 0.000000
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9.3) In E-views, generate a new variable log_GNP (In E-views: Menu Quick, Generate series,
log_GNP=log (GNPDEF).
• Note: A simple but often effective way to stabilize the variance across time is to apply a power
transformation (square root, cube root, log) to the time series. This is the ideal way before using
more sophisticated methods.
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9.4) Plot the graph of the log_GNP series. What can you conclude?
• Once again, the series seems to have an upward trend, and therefore, no constant mean and
variance, suggesting that it is non-stationary.
2.4
2.8
3.2
3.6
4.0
4.4
4.8
50 55 60 65 70 75 80 85 90 95 00 05 10 15
LOG_GNP
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9.5) In E-views run the Augmented Dickey-Fuller Test (ADF) for the above series. What can you
conclude?
• Once again the series is non-stationary.
Null Hypothesis: LOG_GNP has a unit root
Exogenous: Constant
Lag Length: 3 (Automatic - based on SIC, maxlag=15)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -0.570340 0.8735
Test critical values: 1% level -3.452911
5% level -2.871367
10% level -2.572078
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(LOG_GNP)
Method: Least Squares
Date: 01/11/21 Time: 13:48
Sample (adjusted): 1948Q1 2019Q4
Included observations: 288 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
LOG_GNP(-1) -0.000171 0.000300 -0.570340 0.5689
D(LOG_GNP(-1)) 0.560739 0.058242 9.627821 0.0000
D(LOG_GNP(-2)) 0.144034 0.066575 2.163483 0.0313
D(LOG_GNP(-3)) 0.130570 0.058078 2.248192 0.0253
C 0.001812 0.001220 1.484842 0.1387
R-squared 0.630569 Mean dependent var 0.007611
Adjusted R-squared 0.625348 S.D. dependent var 0.006103
S.E. of regression 0.003736 Akaike info criterion -8.324485
Sum squared resid 0.003950 Schwarz criterion -8.260892
Log likelihood 1203.726 Hannan-Quinn criter. -8.299000
F-statistic 120.7609 Durbin-Watson stat 1.898292
Prob(F-statistic) 0.000000
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9.6) Generate the log_gnp in first differences. In E-views, select the Menu Quick, generate
series:
• Once again the series is non-stationary.
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9.7) Plot the graph of the first differences of the log_GNP series. What can you conclude?
• The series seems to have a constant mean and variance (i.e. it appears to cross its mean value
frequently). As such, it seems that we are in the presence of a stationary series.
-.02
-.01
.00
.01
.02
.03
.04
50 55 60 65 70 75 80 85 90 95 00 05 10 15
DLOG_GNP
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9.8) In E-views, run the ADF test in first differences. Use the log_GNP series. What can you
conclude?
• These results suggest that we can reject the unit root hypothesis in the first differences of the log
GNP series. The estimated Tau-statistic is more highly significantly negative than even the 1%
critical value.
