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ContentsIntroduction.............................................................................................................................................. 1
Section 1: Descriptive Analysis ............................................................................................................... 2
1.1 Visual Analysis and Unit Root Tests ............................................................................................. 2
1.2 ARIMA Models ............................................................................................................................... 4
1.2.2: Box-Jenkins Identification ...................................................................................................... 4
1.2.2 Model Selection ...................................................................................................................... 7
1.2.3 Forecast Evaluation and Residual Diagnostics .................................................................... 11
Section 2: Robustness Check ............................................................................................................... 15
2.1 Pretesting .................................................................................................................................... 15
2.2 Correlations ................................................................................................................................. 18
References ............................................................................................................................................ 22
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Introduction
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Section 1: Descriptive Analysis
1.1 Visual Analysis and Unit Root Tests
Figure 1 displays the SAPU index in levels for the full sample; 1994m1 to 2014m6. Several features are
notable here. Firstly, there are several holes in
the data, starting from 2002m3 and ending in
2002m12. As discussed in McLean (2015:***)
these holes are due to a sudden drop in the
volume of news captured by SAMedia over this
period – the reason for this sudden sparseness
has not been ascertained. This data feature
effectively divides the index into two periods; the
first runs from 1994m1 to 2002m2, the second
from 2003m1 to 2014m6.
Secondly, the behaviour of the series appears to
differ markedly between these two sample
periods. As can be seen in Figure 2, the period
1994m1 to 2002m2 appears to be characterized
by a constant mean and variance. The series
does appear to exhibit some degree of cyclicality
over this period, but an Augmented Dickey-
Fuller (ADF) test indicates that we can reject the
hypothesis that this series is I(1) at the one
percent level of significance (the test was
specified with a constant; SIC automatic lag selection selected zero lags – see Table 1).
In contrast with Figure 2, a visual inspection of Figure 3 suggests that the mean and variance of the
data generating process (DGP) appears to increase as a function of time over the period 2003m1 to
2014m6. This portion of the index thus appears to be nonstationary. As is common practice, the
observed increase in the variance of SAPU is
addressed by taking the log of the series (****)
– as can be seen in Figure 4, this course of
action ostensibly reduces the high variance
observed in the latter half of the series. For this
reason, L_SAPU, the log transformed SAPU
index, is used throughout the remainder of this
section.
Regarding the apparent nonstationarity of theseries, Table 2 displays ADF test results for the
0
20
40
60
80
100
120
140
1994 1995 1996 1997 1998 1999 2000 2001
Figure 2: SAPU 1994m1 to 2002m2
0
50
100
150
200
250
300
1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
Figure 1: SAPU 1994m1 to 2014m6
0
50
100
150
200
250
300
2004 2006 2008 2010 2012 2014
Figrue 3: SAPU 2003m1 to 2014m6
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L_SAPU 2003m1 to 2014m6 in levels. The test is specified with a constant and 3 lags. As indicated by
the p-value of 0.4011, we cannot reject the hypothesis that this portion of the series is characterized by
a unit root for any traditional level of significance. This test thus corroborates the hypothesis that the
series is nonstationary.
Regarding the nature of this apparent
nonstationarity, if one simply looks at the graph
of the series it is difficult to tell whether a trend
or a unit root accounts for the nonstationaity; the
series does appear to be characterized by a
slight degree of cyclicality, but it also appears to
trend upward quite consistently. ADF tests also
provide little guidance here: Table 3 shows that
for an ADF test on the level of the series
specified with a constant and a trend we can
reject the hypothesis that the series is nonstationary at the one percent level of significance; as can be
seen in Table 4, the same result is achieve for an ADF test on a constant and the first difference of the
series.
Table 1: ADF Test, 1994m1 to 2002m2 (Levels)
Null Hypothesis: SAPU has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=11)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic: -5.577842 0.0000
Test critical values: 1% level -3.499167
5% level -2.891550
10% level -2.582846
*MacKinnon (1996) one-sided p-values.
.0
.5
4.0
4.5
.0
.5
.0
2004 2006 2008 2010 2012 2014
Figure 4: L_SAPU 2003m1 to 2014m6
Table 2: ADF Test, 2003m1 to 2014m6 (Levels)Null Hypothesis: L_SAPU has a unit root
Exogenous: Constant
Lag Length: 3 (Automatic - based on SIC, maxlag=13)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.755684 0.4011
Test critical values: 1% level -3.479656
5% level -2.883073
10% level -2.578331
*MacKinnon (1996) one-sided p-values.
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1.2 ARIMA Models
Given the findings discussed above, common practice dictates that SAPU is trend stationary (****).
However, economic reasoning might motivate us to treat the series as difference stationary. To
elaborate, one would not expect the DGP of the series to be characterized by a deterministic trend, but
rather by the stochastic development of social and political events. Thus, while a deterministic trend
might fit well with the SAPU sample observed over 2003m1 to 2014m6 in-sample, one would expect
that modelling SAPU with a deterministic trend would result in poor foresting and out-of-sample-fit. This
section tests this hypothesis by estimating and evaluating a battery of ARIMA(p,d,q) models.
1.2.2: Box Jenkins Identification
In following the Box-Jenkins (1976) estimation strategy, this process is initiated with a visual and
statistical inspection of the index, its autocorrelation function (ACF) and its partial autocorrelation
function (PACF). Table 5 presents descriptive statistics for L_SAPU (column 5.1) and two
transformations of L_SAPU: D(L_SAPU) (column 5.2) and DT(L_SAPU) (column 5.3), the first
difference of the log of SAPU and the detrended log of SAPU respectively. For D(L_SAPU) and
DT(L_SAPU), the Jarque-Bera statistic indicates that we can reject the hypothesis that these series are
normally distributed at the one percent level of significance. The third and fourth central moment provide
an indication of the source of this non-normality. The third central moment indicates that D(L_SAPU) is
Table 4: ADF Test, 2003m1 to 2014m6 (1st Difference)
Null Hypothesis: D(L_SAPU) has a unit root
Exogenous: Constant
Lag Length: 2 (Automatic - based on SIC, maxlag=13)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -13.00195 0.0000
Test critical values: 1% level -3.479656
5% level -2.883073
10% level -2.578331
*MacKinnon (1996) one-sided p-values.
Table 3: ADF Test, 2003m1 to 2014m6 (Levels)
Null Hypothesis: L_SAPU has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic - based on SIC, maxlag=13)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -9.393356 0.0000
Test critical values: 1% level -4.026429
5% level -3.442955
10% level -3.146165
*MacKinnon (1996) one-sided p-values.
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skewed right and that DT(L_SAPU) is skewed
left; skewness is most pronounced in
DT(L_SAPU). The fourth central moment
indicates that both series are leptokurtic, with
D(L_SAPU) displaying fatter tails thanDT(L_SAPU). In sum, these test statistics
indicate that D(L_SAPU) and DT(L_SAPU) are
both characterized by a high proportion of data
residing in the tails of their respective
distributions, that D(L_SAPU) has a long right
tail, and that DT(L_SAPU) has a long left tail.
A visual inspection of these series provides an
indication of the source of the non-normality of
D(L_SAPU) and DT(L_SAPU). Referring back to
Figure 4, there appears to be a large downward
spike in the index at 2005m1 and 2003m2.
Column 5.1 in Table 5 indicates that L_SAPU has
an estimated standard deviation of approximately
0.42 and a mean of approximately 4.75;
observation 2005m1, which has a value of 3.26,
is 1.49 units below the mean of the series, and is
exactly one unit below observation 2004m12 and is 1.38 units below observation 2004m2; this
observation thus lies more than two standard deviations below the mean of the series and away from
its neighbouring observations. Similarly, observation 2003m2 (3.49 units) is 0.78 units (less than two
standard deviations) below observation 2003m1, and is 1.06 and 1.26 units below observation 2003m3
and the mean of L_SAPU respectively (both more than two standard deviations).
As can be seen in Figure 5 and Figure
6, differencing and detrending
L_SAPU further exacerbates the
irregularity of these observations,
particularly observation 2005m1.
Figure 5 shows that differencing
L_SAPU causes the deviation in
2005m1 to affect observation
2005m2; 2005m1, with a value of
-1.01, is just less than three standard
deviations below the mean of the
series, and 2005m2, with a value of1.38, is approximately four standard
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2004 2006 2008 2010 2012 2014
Figure 5: D(L_SAPU)
-1.2
-0.8
-0.4
0.0
0.4
0.8
2004 2006 2008 2010 2012 2014
Figure 6: DT(L_SAPU)
Table 5: Descriptive Statistics for L_SAPU Transformations
5.1 5.2 5.3
L_SAPU D(L_SAPU) DT(L_SAPU)
Mean 4.746310 0.007362 1.03E-16
Median 4.693905 -0.012206 -0.008638
Maximum 5.644762 1.383108 0.612533
Minimum 3.255166 -1.006805 -1.142621
Std. Dev. 0.415366 0.345102 0.272905
Skewness -0.286317 0.357227 -0.592941
Kurtosis 3.365061 4.573799 4.478033
Jarque-Bera 2.651773 17.05243 20.64765
Probability 0.265567 0.000198 0.000033
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deviations above the mean of the series. 2003m3 in
Figure 5 (with a value of 1.05) is also approximately
three standard deviations from the mean of the
series. These observations appear to account for the
rightward skewness and thick tails indicated in 5.2.For DT(L_SAPU), observations 2005m1 and
2003m2 are respectively 1.14 and 0.72 units (more
than three and more than two standard deviations)
below the mean of the series, thus accounting for the
leftward skewness and thick tails indicated in 5.3.
To account for the irregularity of these observations,
we restrict the sample to exclude all dates prior to
2003m3 and add dummy variables for observation
2005m1 and 2005m2 to D(L_SAPU) and a dummy variable for observation 2005m1 to DT(L_SAPU).
Table 6 shows descriptive statistics for the residuals of D(L_SAPU) regressed on a constant and these
dummies (6.1) and for DT(L_SAPU) inclusive of the 2005m1 dummy (6.2). For both series, the Jarque-
Bare statistic indicates that the hypothesis of normality cannot be rejected at any traditional level of
significance. All subsequent models presented in this section include the relevant dummy variables and
are estimated on samples which exclude 2003m2. This resolves the issue of outliers.
Turning now to the issue of lag length selection, Figure 7 and 8 respectively display the ACFs and
PACFs for L_SAPU, D(L_SAPU) and DT(L_SAPU). The ACF for L_SAPU shows a markedly slow linear
decay, a feature indicative of a unit root (Enders, 2010:73); however, the ACFs for D(L_SAPU) and
DT(L_SAPU) show that either approach removes this persistence, reducing the number of significant
Table 6: Accounting for Outliers
6.1 6.2
D(L_SAPU) DT(L_SAPU)
Mean -1.76E-17 2.35E-16
Median -0.008600 -0.015972
Maximum 0.695292 0.602350
Minimum -0.748037 -0.624201
Std. Dev. 0.294703 0.246021
Skewness -0.010816 -0.037858
Kurtosis 2.558378 2.755471
Jarque-Bera 1.091530 0.365859
Probability 0.579398 0.832827
-0,600
-0,400
-0,200
0,000
0,200
0,400
0,600
0,800
1 2 3 4 5 6 7 8 9 10
Figure 7: SAPU Autocorrelation Function
L_SAPU D(L_SAPU) DT(L_SAPU) Significance
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lags to one in both cases. Turning to Figure 8, the PACF of D(L_SAPU) indicates that our ARIMA model
may need to contain as many as three lags, while the PACF of DT(L_SAPU) indicates that one lag may
be sufficient.
1.2.2 Model Selection
Given the ambiguity of these indicators, the model selection process employed here proceeds as
follows: Two sets of models are generated here; a set of ARIMA(p,1,q) models (i.e. a set with
D(L_SAPU) as the dependant variable) all specified with a constant, and a set of ARIMA(p,0,q) models
(i.e. a set with L_SAPU as the dependant variable) all specified with a constant and a linear time trend.
In consideration of the PACF of D(L_SAPU), models with up to three lags are considered; all models
are specified to contain either an AR or an MA term at every lag length lower than the model’s longest
lag. Carrying out the above yield two sets of fourteen models.
Estimation of these models is carried out over a sample period starting no earlier than 2003m5 and
running up to 2012m6. In addition to omitting the aforementioned outlier, the lower bound of each
estimate was selected so that all models are estimated on the same number of observations; this is
imperative, as it is under these conditions that the Akaike Information Criterion (AIC) and Schwartz
Bayesian Criterion (SBC) allow us to compare the goodness of fit for non-nested models (Enders,
2010:71). The upper bound was selected so as to conserve a sample period (2012m7 to 2014m6, 23
observations in total) for out-of-sample model evaluation.
The set of ARIMA(p,1,q) models are displayed in Table 7.1 and Table 7.2, and the set of ARIMA(p,0,q)
models are displayed in Table 8.1 and Table 8.2. Statistical significance is indicated by asterisks: ***,
**, * indicate statistical significance at the one, five and ten percent level respectively. Notably, the
-0,600
-0,400
-0,200
0,000
0,200
0,400
0,600
0,800
1 2 3 4 5 6 7 8 9 10
Figure 8: SAPU Partial Autocorrelation Function
L_SAPU D(L_SAPU) DT(L_SAPU) Significance
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highest adjusted R-squared (of 0.511053, column 7.9) achieved by the set of ARIMA(p,1,q) models is
less than the lowest adjusted R-squared (of 0.564614, column 8.2) achieved by the set of ARIMA(q,0,1)
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Table 7.1: L_SAPU ARIMA(p,1,q) Modes (Specified with a Constant)
7.1 7.2 7.3 7.4 7.5 7.6 7.7
AR(1) -0.442928*** -0.563887*** -0.799862*** -0.657519***
AR(2) -0.300189*** -0.079993 -0.450292***
AR(3) -0.264077***
MA(1) -0.811335*** -0.773702*** -0.801019***
MA(2) -0.652691*** -0.179567*
MA(3)
R-Squared 0.378499 0.503519 0.440800 0.503985 0.505632 0.515960 0.483596
Adjusted R-Squared 0.360910 0.489467 0.419497 0.485089 0.486799 0.497521 0.458769
AIC 0.262895 0.038304 0.175446 0.055546 0.052221 0.031107 0.114011
SBC 0.361094 0.136503 0.298196 0.178295 0.174970 0.153856 0.261310
Table 7.2: L_SAPU ARIMA(p,1,q) Models (Specified with a Constant)
7.8 7.9 7.10 7.11 7.12 7.13 7.14
AR(1) -0.689573*** -0.759654*** -0.749900*** AR(2) -0.500107*** -0.133164 -0.093440 AR(3) 0.068400 -0.085062 -0.046456MA(1) -0.784627*** -0.764419*** -0.809714*** -0.754525***MA(2) -0.738372*** -0.094265 -0.170669* -0.651971***MA(3) -0.412994*** -0.243409*** -0.100864 -0.213780**
R-Squared 0.502941 0.533482 0.521552 0.523528 0.518376 0.509577 0.506587 Adjusted R-Squared 0.479044 0.511053 0.498549 0.500621 0.495221 0.485999 0.482865 AIC 0.075830 0.012419 0.037670 0.033530 0.044286 0.062389 0.068468SBC 0.223129 0.159718 0.184969 0.180829 0.191585 0.209688 0.215767
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Table 8.1: L_SAPU ARIMA(p,0,q) Models (Specified with a Constant and a Linear Time Trend)
8.1 8.2 8.3 8.4 8.5 8.6 8.7
AR(1) 0.281013*** 0.249483** 0.250299** 0.233942** AR(2) 0.124413 0.190037* 0.098348 AR(3) 0.112301MA(1) 0.242873** 0.277499*** 0.265961***MA(2) 0.092901 0.144951MA(3)
R-Squared 0.580671 0.576597 0.591906 0.584635 0.591224 0.586793 0.593339 Adjusted R-Squared 0.568803 0.564614 0.572286 0.568811 0.575651 0.571052 0.573788 AIC 0.026632 0.036300 0.035837 0.035316 0.019326 0.030105 0.032320SBC 0.124831 0.134499 0.183136 0.158065 0.142075 0.152854 0.179619
Table 8.2: DT(L_SAPU) ARIMA(p,0,q)
Models (Specified with a Constant and a Linear Time Trend
8.8 8.9 8.10 8.11 8.12 8.13 8.14
AR(1) 0.239662** 0.242073** 0.247522** AR(2) 0.125431 0.189134* 0.182591* AR(3) 0.106804 0.125459 0.089720MA(1) 0.262686*** 0.275539*** 0.261801*** 0.256385**MA(2)
0.088107 0.146813 0.152479 0.076966MA(3) 0.009099 0.039288 0.041488 0.006766
R-Squared 0.587568 0.585234 0.587626 0.591246 0.592057 0.592257 0.594761 Adjusted R-Squared 0.567739 0.565293 0.567800 0.571594 0.572444 0.572653 0.575278 AIC 0.046411 0.052054 0.046270 0.037454 0.035468 0.034978 0.028817SBC 0.193710 0.199353 0.193569 0.184753 0.182767 0.182277 0.176116
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models. This finding corroborates the first element of our hypothesis that models including a linear trend
will fit well in-sample. Note however that adjusted R-squared is not an appropriate criterion for
evaluating the relative goodness of fit for non-nested models (Wooldridge, *****); as mentioned above,
for this purpose we must refer to each model’s AIC and SBC score . Moreover, it is also necessary to
note that the AIC and SBC cannot be used to rank models between these groups, as they do not allowfor comparison across models with different transformations of the dependent variable (Burnham &
Anderson, 2002:80). Rather, the AIC and he SBC are used here to select the best models from within
each group respectively.
For ease of evaluation, the four lowest (and thus best) AIC and SBC scores among each of the two sets
of models have been colour-coded: light blue is lowest, dark blue is second lowest, light gold is third
lowest, dark gold is fourth lowest. Of the ARIMA(p,1,q) models, the ARIMA(0,1,2) model in column 7.6
scores lowest in both AIC and SBC and is thus the strongest contender of this group; the ARIMA(1,1,2)
model (7.9) is also a strong contender, with the second lowest AIC score and the third lowest SBC
score. For the third ARIMA(p,1,q) contender, the ARIMA(0,1,1) model (7.2) is selected given that it
attains the second lowest SBC score; this statistic is unbiased in small samples (****), and thus 7.2 is
selected not only for the level of its SBC score but also for the reliability of this score vis-à-vis models
that achieved low AIC scores.
Among the ARIMA(p,0,q) models, the ARIMA(1,0,0) model (8.1) achieves the lowest SBC score and
the second lowest AIC score, and is thus selected. The ARIMA(0,0,1) model (8.2) performs poorly
relative to many of the other ARIMA(p,0,q) models with respects to its AIC score, but is selected on the
basis of its SBC score (which is second-lowest among this group of models). Finally, the ARIMA(1,0,1)
model (8.5), which achieved the lowest AIC score and the third lowest SBC score, is selected.
1.2.3 Forecast Evaluation and Residual Diagnostics
Table 9 presents forecast evaluation statistics for the six models selected above. Let us first consider
the root mean squared error (RMSE), the mean absolute error (MAE) and the mean absolute
percentage error (MAPE). The formulas for these statistics as calculated by EViews (EViews User’s
Guide Part II, “Basic Data Analysis”, 2015) are given as follows:
√ ∑ (∗ − )2ℎ+=+ [1]
∑ |∗ − |ℎ+=+ [2]
100 ∑ ∗ −
+
=+ /ℎ [3]
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where the ∗ are the forecasted values of series, the are the actual values of the series, ℎ is the
number of observations that comprise the forecast period and is the final period of the sample used
for estimation. It can be deduced from the equations above that the RMSE and the MAE are invariant
to additive transformations to
{} and
{∗
} (the series of
and
∗ respectively) and are sensitive to
multiplicative transformations thereof;
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Table 9: Forecast Evaluation Statistics
7.2 7.6 7.9 8.1 8.2 8.5
Root Mean Squared Error 0,233025 0,233939 0,232661 0,227842 0,222692 0,241225
Mean Absolute Error 0,174137 0,177961 0,175856 0,176699 0,170752 0,188668
Mean Abs, Percent Error 81,88688 114,4257 119,8167 3,423309 3,303542 3,657336
Theil Inequality Coefficient (UI) 0,490039 0,478561 0,497255 0,021892 0,02141 0,023171
Bias Proportion 0,038518 0,045005 0,006261 0,013418 0,021025 0,009302
Variance Proportion 0,179617 0,135159 0,191533 0,467837 0,513652 0,379572
Covariance Proportion 0,781866 0,819836 0,802206 0,518745 0,465323 0,611126
Theil Inequality Coefficient (UII) 0,819317 0,822532 0,818037 0,043657 0,042670 0,046221
Table 10: Residual Diagnostics
7.2 7.6 7.9 8.1 8.2 8.5
Jarque-Bera Test for Normality
Jarque-Bera Statistic 1,131019 0,703254 0,676746 0,980092 0,926902 0,667017
Prob, 0,568071 0,703542 0,712929 0,612598 0,629109 0,716406
Breusch-Godfry Serial Correlation Test, 4 Lags
F-statistic 0,471431 1,804706 0,406063 1,26157 2,001406 0,996703
Prob. 0,7566 0,1337 0,8039 0,29 0,0999 0,413
Obs*R-squared 1,956233 6,898901 1,59509 5,185523 8,004894 4,176669
Prob. 0,7438 0,1413 0,8097 0,2688 0,0914 0,3826
ARCH Heteroskedasticity Test, One Lag
F-statistic 0,105423 0,067843 0,157677 0,087633 0,055419 0,204926
Prob. 0,746 0,795 0,6921 0,7678 0,8143 0,6517
Obs*R-squared 0,107288 0,069067 0,160388 0,089198 0,056425 0,208357
Prob. 0,7433 0,7927 0,6888 0,7652 0,8122 0,6481
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the opposite is true of the MAPE. However, none of these three statistics are comparable across
difference transformations of the dependent variable, as the series {} and {∆} do not generally
contain the same forecastable information (*****).
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Section 2: Robustness Check
As a robustness check, McLean (2015) evaluates the relationship between the L_SAPU index and
L_SAVI (the log of the South African Volatility Index) over the period 2007m7 to 2012m11. The results
found here were mixed. Correlation between L_SAVI and L_SAPU was found to be negligible at 0.013,
but was also found to be much higher for the first differences of these series (0.177); this finding
suggests that there is non-negligible correlation between the changes in these indices (Mclean,
2015:14-15). Less positively, a simple regression of the L_SAVI on L_SAPU did not produce a
statistically significant coefficient, a result that is probably partially a product of the small sample of 65
observations used in these estimates, but which nevertheless bodes ill as a reflection of the robustness
of the index (McLean, 2015: 15).
In light of the opacity of this previous investigation, this section adds to McLean’s (2015) robustness
checks with an examination of the relationship between SAPU and an index (administered by the BER
(Bureau for Economic Research) and obtained from Quantec’s Easy Data (2015) database) which
tracks the percentage of a representative sample of manufacturing firms that cite the current political
climate as a constraint on production. This index is denoted here as MSPC.
Thinking about the data generating process underlying both of these indices leads to two testable
hypotheses. Firstly, the theoretical and empirical literature documenting the effects of policy uncertainty
on production suggest that policy uncertainty constrains production by increasing the option value of
future investment (Rodrik, 1991); thus, one would expect SAPU and MSPC to be positively correlated
and possibly cointegrated. Secondly, SAPU is arguably a direct measure of policy uncertainty, whileMSPC is a variable that should respond to policy uncertainty; thus one would expect SAPU to be weakly
exogenous with respects to MSPC.
2.1 Pretesting
Table 12 shows the ADF critical and test statistics for a variety of transformations on MSPC and under
a variety of specifications, broken into sample periods that correspond with the data availability of
SAPU. As indicated in 12.1 and 12.2 respectively, ADF tests conducted over the full sample, specified
with a constant or a constant and a deterministic trend, produce test statistics that are too high to reject
the hypothesis of a unit root; high p-values of 0.5963 and 0.8504 respectively make this rejection
uncontentious. Furthermore, under 12.3 it can be seen that for the first difference of MSPC (D_MSPC)
the hypothesis of a unit root can be rejected at the one percent level of significance; hence we can
conclude from these full sample tests that MSPC is difference stationary (i.e. is I(1)). With reference to
12.10, 12.11 and 12.12, the same conclusion may be drawn with regards to the log of MSPC (L_MSPC)
and for the first difference of the log of MSPC (DL_MSPC).
When MSPC is examined in sections corresponding the data availability of SAPU, tests for the level of
integration of the series produce results similar to those reported in Section 1 for SAPU in the
corresponding sample periods: For the period 1994Q1 to 2001Q4, MSPC and L_MSPC tests as I(0);
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Table 12: ADF tests MSPC
1994Q1 to 2014Q2 1994Q1 to 2001Q4 2003Q1 to 2014Q2
12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9
Dependent Variable MSPC MSPC D(MSPC) MSPC MSPC D(MSPC) MSPC MSPC D(MSPC)
t-Statistic -1.363019 -1.411030 -13.40636 -5.061119 -5.382322 -9.542237 -0.944454 -3.641963 -9.035099
ADF Test critical values:
1% level -3.514426 -4.076860 -3.514426 -3.661661 -4.284580 -3.670170 -3.581152 -4.170583 -3.581152
5% level -2.898145 -3.466966 -2.898145 -2.960411 -3.562882 -2.963972 -2.926622 -3.510740 -2.92662210 % level -2.586351 -3.160198 -2.586351 -2.619160 -3.215267 -2.621007 -2.601424 -3.185512 -2.601424
Prob.* 0.5963 0.8504 0.0001 0.0003 0.0007 0.0000 0.7649 0.0371 0.0000
Exogenous: Constant Constant Constant Constant Constant Constant Constant Constant Constant
Trend Trend Trend
12.10 12.11 12.13 12.14 12.15 12.16 12.17 12.8 12.9
Dependent Variable L_MSPC L_MSPC D(L_MSPC) L_MSPC L_MSPC D(L_MSPC) L_MSPC L_MSPC D(L_MSPC)
t-Statistic -2.313132 -2.312348 -12.56348 -4.813128 -5.105216 -9.568363 -0.992759 -3.538037 -8.323042
ADF Test critical values:
1% level -3.513344 -4.075340 -3.514426 -3.661661 -4.284580 -3.670170 -3.581152 -4.170583 -3.581152
5% level -2.897678 -3.466248 -2.898145 -2.960411 -3.562882 -2.963972 -2.926622 -3.510740 -2.926622
10 % level -2.586103 -3.159780 -2.586351 -2.619160 -3.215267 -2.621007 -2.601424 -3.185512 -2.601424
Prob.* 0.1704 0.4224 0.0001 0.0005 0.0013 0.0000 0.7482 0.0470 0.0000
Exogenous: Constant Constant Constant Constant Constant Constant Constant Constant Constant
Trend Trend Trend
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for the period 2003Q1 to 2014Q2, MSPC and L_MSPC test as trend stationary at the five percent level
of significance.
Table 13 shows the results of a similarly implemented battery of ADF tests for a variety of
transformations of SAPU; as in Section 1, full sample tests are omitted here due to the discontinuity in
the data from 2002Q1 to 2002Q4. For the period 2003Q1 to 2014Q2 the results in Table 13 are similar
to those presented in Section 1: 13.4 and 13.10 indicate that we cannot reject the hypothesis that SAPU
and L_SAPU are nonstationary, while the rejection of the hypothesis in 13.5 and 13.11 suggest that the
data is trend stationary. For 1994Q1 to 2001Q4, the results obtained for SAPU differ importantly from
those obtained for the series’ monthly counterpart: for this period and at this data frequency, the
hypothesis that the series is non-stationary cannot be rejected at the ten percent level for SAPU and
L_SAPU; the rejection of the null hypothesis obtained in 13.2 and 13.5 thus suggest that this portion of
the series is trend stationary.
Table 13: ADF Tests SAPU (Quarterly)
1994Q1 to 2001Q4 2003Q1 to 2014Q2
13.1 13.2 13.3 13.4 13.5 13.6
Dependent Variable SAPU SAPU D(SAPU) SAPU SAPU D(SAPU)t-Statistic -2.592892 -3.791835 -5.627037 -2.424996 -3.959771 -8.359628
ADF Test critical values:
1% level -3.689194 -4.323979 -3.711457 -3.581152 -4.170583 -3.581152
5% level -2.971853 -3.580623 -2.981038 -2.926622 -3.510740 -2.926622
10 % level -2.625121 -3.225334 -2.629906 -2.601424 -3.185512 -2.601424
Prob.* 0.1063 0.0323 0.0001 0.1407 0.0172 0.0000
Exogenous: Constant Constant Constant Constant Constant Constant
Trend Trend
13.7 13.8 13.9 13.10 13.11 13.12
Dependent Variable L_SAPU L_SAPU D(L_SAPU) L_SAPU L_SAPU D(L_SAPU)
t-Statistic -2.432925 -5.319496 -4.917714 -1.313868 -3.931164 -9.590602
ADF Test critical values:
1% level -3.661661 -4.394309 -3.752946 -3.588509 -4.175640 -3.588509
5% level -2.960411 -3.612199 -2.998064 -2.929734 -3.513075 -2.929734
10 % level -2.619160 -3.243079 -2.638752 -2.603064 -3.186854 -2.603064
Prob.* 0.1414 0.0013 0.0007 0.6149 0.0186 0.0000
Exogenous Constant Constant Constant Constant Constant Constant
Trend Trend
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2.2 Correlations
Though it is not robust to sources of endogeneity, the correlation structure between variables can
provide valuable insight regarding the relationship between them. Table 14 displays the correlation
coefficients for the SAPU and MSPC in levels, first differences, log levels and for the first differences of
the log levels of these series. Columns corresponds to lags, leads or contemporaneous value of SAPU
or its transformation (the relevant transformation of SAPU and MSPC for each row is indicated in the
left hand column); a positive (negative) number indicates that the reported correlation is between MSPC
and a lead (lag) of SAPU (or transformations thereof). For convenience, positive correlations are
highlighted in yellow, negative correlations are highlighted in blue.
As can be seen in Table 14 below, correlation between MSPC and SAPU for the period 1994Q1 to
2001Q4 (14.1) is positive for contemporaneous values of SAPU, as well as for its first and second lag.
Leads of SAPU are found to be negatively correlated with MSPC, but these correlations are of a
negligible magnitude. This correlation structure is consistent with the above-stated hypotheses that
SAPU and MSPC should be positively correlated and that SAPU should be seen to drive the variation
in MSPC. However, this result is somewhat reversed for L_MSPC and L_SAPU: here, leads of L_SAPU
are found to be positively correlated with L_MSPC, and lags of L_SAPU are negligibly negatively
correlated with L_MSPC, suggesting that it is L_MSPC that drives L_SAPU. Furthermore, for D(MSPC)
Table 14: Correlations for Leads and Lags of SAPU
14.1: 1994Q1 to 2001Q4
+2 +1 0 -1 -2
MSPC and SAPU -0,0073 -0,0141 0,1821 0,2347 0,2393
D(MSPC) and D(SAPU) 0,1435 -0,1465 0,1907 0,0236 0,0526
L_MSPC and L_SAPU 0,3056 0,2607 0,2506 -0,0400 -0,0789
D(L_MSPC) and D(L_SAPU) -0,0014 -0,1424 0,2917 -0,0257 0,1232
14.2: 2003Q1 to 2014Q2
+2 +1 0 -1 -2
MSPC and SAPU 0,7687 0,8255 0,8853 0,9050 0,8844
D(MSPC) and D(SAPU) -0,1361 -0,0691 0,2428 0,1166 0,0842
L_MSPC and L_SAPU 0,7987 0,8424 0,8771 0,8817 0,8661
D(L_MSPC) and D(L_SAPU) 0,0527 0,0230 0,2459 -0,0380 -0,1001
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and D(SAPU) and for D(L_MSPC) and
D(L_SAPU), the correlation structure for lags
and leads of SAPU breaks down into obscure
patterns. However, there remains the positive
finding that for all transformations of the dataSAPU and MSPC are contemporaneously
positively correlated; this is the least one
would expect if these two series contain
information derived from the same underlying
source.
Referring now to 14.2, the correlation structure
between SAPU and MSPC is very pronounced
in the period 2003Q1 TO 2014Q2, with
positive correlations ranging from a low of
0,7687 at two leads of SAPU, up to a high of
0,9050 at one lag of SAPU; similarly
pronounced correlations are also evident for
L_SAPU and L_MSPC. Little of a conclusive
nature can be drawn from this association
given the indications of nonstationarity that
characterize this section of the data (as
discussed in Section 2.1), but nevertheless
this result is very encouraging. To put this
finding into perspective, Baker, Bloom and
Davis (2013:18), in checking the robustness of
their United States policy uncertainty index
(USPU), found that the VIX and USPU shared
a contemporaneous correlation of 0.578. For
a more direct check of the efficacy of their
methodology, Baker, Bloom and Davis(2013:18) also construct a news-based equity
market uncertainty (EMU) index and find that it exhibits a contemporaneous correlation of 0.733 with
the VIX. Though Baker, Bloom and Davis (2013) do not report on the stationarity of these series, a
visual inspection of them (see Baker, Bloom and Davis (2013:46)) suggests that they are also
nonstationary. Given this context, the findings presented here, which indicate a greater
contemporaneous correlation between SAPU (L_SAPU) and MSPC (L_MSPC) than was found
between the VIX and Baker, Bloom and Davis’ (2013) EMU index, should be regarded as strong
evidence of the reliability of SAPU as a measure of policy uncertainty.
2
3
4
5
6
7
94 96 98 00 02 04 06 08 10 12 14
L_SAPU L_MSPC
Figure 9: L_SAPU and L_MSPC 1994Q1 to 2014Q2
-1.6
-1.2
-0.8
-0.4
0.0
0.4
0.8
1994 1995 1996 1997 1998 1999 2000 2001
L_SAPU L_MSPC
Figure 10: L_SAPU and L_MSPC 1994Q1 to 2001Q4 (Demeaned)
-1.2
-0.8
-0.4
0.0
0.4
0.8
03 04 05 06 07 08 09 10 11 12 13 14
L_SAPU L_MSPC
Figure 11: L_SAPU and L_MSPC 2003Q1 to 2014Q2 (Demeaned)
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Bonate, P. 2006 Pharmacokinetic-Pharmacodynamic Modeling and Simulation. New York: Springer
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Burnham, K.P. & Anderson, D.R. 2002. Model Selection and Multimodel Inference: A Practical
Information-Theoretic Approach. New York: Springer Inc.