Estimating an ARDL model: best practices and dynamac
dynamac includes two datasets. We will use the Wright (2017) dataset on income inequality. We can look at the first few rows of this dataset
library(dynamac)
head(ineq)
#> year mood urate concern demcontrol incshare10 csentiment incshare01
#> 1 1966 60.793 3.8 0.465 1 0.3198 97.0 0.0837
#> 2 1967 61.332 3.8 0.475 1 0.3205 94.7 0.0843
#> 3 1968 58.163 3.6 0.490 1 0.3198 94.2 0.0835
#> 4 1969 54.380 3.5 0.507 0 0.3182 90.3 0.0802
#> 5 1970 60.555 4.9 0.519 0 0.3151 75.8 0.0780
#> 6 1971 64.077 5.9 0.531 0 0.3175 80.6 0.0779concern is public concern about income inequality;
incshare10 is the income share of the top ten percent;
urate is the unemployment rate. Wright (2017) argues that concern about income
inequality grows as inequality itself worsens and economic conditions
deteriorate. A simple model, then, is that concern is a function of past
values of itself, the level of income share held by the top ten percent,
changing levels of income share held by the top ten percent, as
well changing levels of unemployment in the short term.
\[\Delta Concern_t = \alpha_0 + \phi_1 Concern_{t-1} + \theta_1 Income Top 10_{t-1} + \beta_1 \Delta Income Top 10_t + \beta_2 \Delta Unemployment_t + \epsilon_t\]
where the residuals are white noise. Let’s develop the corresponding ARDL model using dynamac.
Understanding our time series
Step 1 in any time series analysis is a visual inspection of the series coupled with formal tests of stationarity: whether the series has a constant mean, variance, and covariance over time (so that it reverts back to mean level), or if the series instead violates any of these three conditions. We advocate for using the urca package for this (Pfaff, Zivot, and Stigler 2016), which includes a variety of tools and tests. Simply plotting the series reveals the following:
library(urca)
ts.plot(ineq$concern)
ts.plot(ineq$incshare10)
ts.plot(ineq$urate)
None of the three series looks especially well-behaved.
concern rose quickly and has been moving sluggishly since.
incshare10 has only grown over time, which cannot be
mean-reverting (like a stationary series). urate
experiences steep peaks and valleys with significant interludes in
between. All three series look integrated. But we should be more
discerning by using empirical tests, also included in urca. So we can
execute the Augmented Dickey-Fuller (ADF) test, Phillips-Perron test,
DF-GLS test, and KPSS test on each series. On concern this
looks like
summary(ur.df(ineq$concern, type = c("none"), lags = 1))
#>
#> ###############################################
#> # Augmented Dickey-Fuller Test Unit Root Test #
#> ###############################################
#>
#> Test regression none
#>
#>
#> Call:
#> lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.029626 -0.006624 0.000511 0.011123 0.033842
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> z.lag.1 0.002474 0.003597 0.688 0.495
#> z.diff.lag -0.131523 0.148125 -0.888 0.379
#>
#> Residual standard error: 0.01422 on 45 degrees of freedom
#> Multiple R-squared: 0.0243, Adjusted R-squared: -0.01907
#> F-statistic: 0.5603 on 2 and 45 DF, p-value: 0.575
#>
#>
#> Value of test-statistic is: 0.6878
#>
#> Critical values for test statistics:
#> 1pct 5pct 10pct
#> tau1 -2.62 -1.95 -1.61
summary(ur.pp(ineq$concern, type = c("Z-tau"), model = c("constant"), use.lag = 1))
#>
#> ##################################
#> # Phillips-Perron Unit Root Test #
#> ##################################
#>
#> Test regression with intercept
#>
#>
#> Call:
#> lm(formula = y ~ y.l1)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.031791 -0.006041 0.000738 0.006481 0.031282
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.09997 0.02946 3.393 0.00143 **
#> y.l1 0.83010 0.05085 16.324 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.01274 on 46 degrees of freedom
#> Multiple R-squared: 0.8528, Adjusted R-squared: 0.8496
#> F-statistic: 266.5 on 1 and 46 DF, p-value: < 2.2e-16
#>
#>
#> Value of test-statistic, type: Z-tau is: -3.4373
#>
#> aux. Z statistics
#> Z-tau-mu 3.496
#>
#> Critical values for Z statistics:
#> 1pct 5pct 10pct
#> critical values -3.571174 -2.92277 -2.599003
summary(ur.ers(ineq$concern, type = c("DF-GLS"), model = c("constant"), lag.max = 1))
#>
#> ###############################################
#> # Elliot, Rothenberg and Stock Unit Root Test #
#> ###############################################
#>
#> Test of type DF-GLS
#> detrending of series with intercept
#>
#>
#> Call:
#> lm(formula = dfgls.form, data = data.dfgls)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.027024 -0.003352 0.003511 0.013156 0.036332
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> yd.lag -0.02951 0.03305 -0.893 0.377
#> yd.diff.lag1 -0.10824 0.14675 -0.738 0.465
#>
#> Residual standard error: 0.01417 on 45 degrees of freedom
#> Multiple R-squared: 0.0312, Adjusted R-squared: -0.01186
#> F-statistic: 0.7247 on 2 and 45 DF, p-value: 0.4901
#>
#>
#> Value of test-statistic is: -0.8928
#>
#> Critical values of DF-GLS are:
#> 1pct 5pct 10pct
#> critical values -2.61 -1.95 -1.62
summary(ur.kpss(ineq$concern, type = c("mu"), use.lag = 1))
#>
#> #######################
#> # KPSS Unit Root Test #
#> #######################
#>
#> Test is of type: mu with 1 lags.
#>
#> Value of test-statistic is: 0.6415
#>
#> Critical value for a significance level of:
#> 10pct 5pct 2.5pct 1pct
#> critical values 0.347 0.463 0.574 0.739The ADF test, Phillips-Perron test, and DF-GLS test all have null
hypotheses of a unit root, all of which we fail to reject. The KPSS test
has a null hypothesis of stationarity which we do reject. We
have compelling evidence, then, of integration (I[1]) in the series
concern. We can check whether differencing is enough to
make the series stationary by executing the same tests
summary(ur.df(diff(ineq$concern), type = c("none"), lags = 1))
summary(ur.pp(diff(ineq$concern), type = c("Z-tau"), model = c("constant"), use.lag = 1))
summary(ur.ers(diff(ineq$concern), type = c("DF-GLS"), model = c("constant"), lag.max = 1))
summary(ur.kpss(diff(ineq$concern), type = c("mu"), use.lag = 1))Each of the above tests, when run, indicated that the differenced
series of concern is stationary. Having gathered the basic
information about the nature of the history of our variables, we might
be itching to estimate some preliminary models. To this point, R hasn’t
offered much in the way of improving beyond the basic lm()
framework for using regression to estimate time series models in a
consistent syntax. We elaborate on this problem and illustrate our
solution, dynardl.
Estimating ARDL models with dynardl
ARDL models are flexible, but their flexibility often results in
variables of different lengths due to differencing and lagging. For
instance, consider our simple model from above where
incshare10 appears as a first lag and a first difference,
urate appears as a first difference, there is a lagged
dependent variable, and concern is the dependent variable
in differences. We can introduce a lag through the built-in
lshift function in dynamac. The syntax is just
lshift(variable, num.lags), where num.lags is
the number of periods for the variable to be lagged. We can also
difference a series through dshift. The syntax is just
dshift(variable) for a first difference. For instance,
head(ineq$incshare10)
#> [1] 0.3198 0.3205 0.3198 0.3182 0.3151 0.3175
head(lshift(ineq$incshare10, 1))
#> [1] NA 0.3198 0.3205 0.3198 0.3182 0.3151
head(dshift(ineq$incshare10))
#> [1] NA 0.0007 -0.0007 -0.0016 -0.0031 0.0024So the syntax for the simple model described would be easy to write. But notice the problem
summary(lm(diff(ineq$concern) ~ lshift(ineq$concern, 1) + lshift(ineq$incshare10, 1) + dshift(ineq$incshare10) + dshift(ineq$urate)))
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'summary': variable lengths differ (found for 'lshift(ineq$concern, 1)')Introducing the lag changed the variable lengths, and we probably
don’t want to have to think about time series operation in our model
specification, anyway. Other software has unified this operation by
introducing a common syntax, like d. for differencing and l. for
lagging, or even l.d. for lag differences (what program could that be?).
In R, though, it remains more of a nuisance than it should be. The
dynardl function helps to remedy this challenge by
encouraging the user only to think about model structure in exactly the
way outlined above: which variables \(x\) get lags \(l\) and lagged differences \(q\), etc. The function expects a basic
formula of all the variables in the model, the data set
(data), if there is one, then lagged levels
(lags) and lagged differences (lagdiffs) as a
list and differences (diffs) and contemporaneous levels
(levels) as a vector. The sole exception is if the user
wants to run the model in differences, s/he needs to specify
ec = TRUE (in an effort to force us to think critically
about the error-correction form of the model). For instance, our above
model now becomes (ignoring the simulate option for
now)
dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
ec = TRUE, simulate = FALSE)Purely hypothetically, if we wanted more lagged levels of
concern, we would just change 1 to
c(1:5) for lags at \(t-1\)
to \(t-5\), or any other number of
lags. If we wanted to include the first-difference of urate
lagged at periods \(t-1\) and \(t-2\), we would now run
dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("urate" = c(1, 2)),
ec = TRUE, simulate = FALSE)If the variable can appear in multiple periods (lags and
lagdiffs), it must be specified as a list. If it can only
appear contemporaneously (levels and diffs),
it must be specified as a vector. (The alternative was to specify
levels through a lag at time 0: this did not
seem practical.) We can also add or suppress a constant from the model
by specifying constant = TRUE/FALSE, and we can do the same
with a linear trend term with trend = TRUE/FALSE (by
default, constants are included and trends are not).
The option ec specifies the nature of the dependent
variable. If it is possibly error-correcting (ec = TRUE),
it is run in differences, or period-over-period changes. If it is not
(ec = FALSE), the dependent variable is run in levels.
At this point, dynardl is just a unifying way of
thinking about time series models for estimation. Yet it is going to
offer two other important advantages: interpreting the effects of our
variables, and executing a powerful test for cointegration. We will
start with the latter. Remember testing for I(1) processes earlier: each
test indicated that concern was I(1). Running the same
tests for each of incshare10 and urate
suggests that all three series are I(1). This might cause us concern
about cointegration: the special relationship between I(1) series where
the series are in a long-run relationship, even if they move apart in
the short term. Cointegrating series are difficult to uncover.
Traditional methods, like the Engle and Granger (1987) two-step method or likelihood-based
approaches (Johansen 1995) too often
conclude cointegration when it does not exist, at least in smaller
sample sizes (Philips 2018). An
alternative test (Pesaran, Shin, and Smith
2001), which we refer to as the ARDL-bounds procedure, performs
much better in small samples (\(t <
80\)), but it is more cumbersome to implement. The package
dynamac is meant to resolve that deficiency by implementing critical
value testing procedure for the user. Following Philips (2018), it requires estimating the relationship
in error-correction form, obtaining statistics of interest, and then
testing them via the function pssbounds.
Cointegration testing using the ARDL-bounds procedure and
pssbounds
The ARDL-bounds procedure begins with two preliminaries. First, we
must ensure that the regressors are not of order I(2) or more and that
any seasonal components have been removed. We demonstrated this above
when we found that the first-difference of incshare10 and
urate were both stationary. In addition, there were no
seasonal components, looking at the series (although more formal
diagnostics are probably warranted). Second, we must ensure that the
dependent variable is integrated of order I(1). And again,
above, we demonstrated that incshare10 is integrated.
The next step in ARDL-bounds is to estimate the error-correction form
of the model. Two points are key: the independent variables that are
potentially I(1) must be entered in levels, and the resulting residuals
of the error-correction ARDL model must be white noise (random with no
residual autocorrelation). Here, that means that we run our dependent
variable in differences, we include the lagged levels of the dependent
variable, we include the levels of the potentially cointegrating
variable, incshare10, and the short-run effects of
urate through differences. Returning to our model above,
and using dynardl, this is now straightforward:
res1 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
ec = TRUE, simulate = FALSE)
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
summary(res1)
#>
#> Call:
#> lm(formula = as.formula(paste(paste(dvnamelist), "~", paste(colnames(IVs),
#> collapse = "+"), collapse = " ")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.025848 -0.005293 0.000692 0.006589 0.031563
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.122043 0.027967 4.364 7.87e-05 ***
#> l.1.concern -0.167655 0.048701 -3.443 0.0013 **
#> d.1.incshare10 0.800585 0.296620 2.699 0.0099 **
#> d.1.urate 0.001118 0.001699 0.658 0.5138
#> l.1.incshare10 -0.068028 0.031834 -2.137 0.0383 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.01169 on 43 degrees of freedom
#> (1 observation deleted due to missingness)
#> Multiple R-squared: 0.3671, Adjusted R-squared: 0.3083
#> F-statistic: 6.236 on 4 and 43 DF, p-value: 0.0004755Next we need to ensure that the residuals from this model are white
noise. To help with this, we also introduce
dynardl.auto.correlated. This expects a
dynardl model and implements a few white-noise-residual
tests with readable output that reminds of the null hypotheses. Here, we
just run
dynardl.auto.correlated(res1)
#>
#> ------------------------------------------------------
#> Breusch-Godfrey LM Test
#> Test statistic: 3.704
#> p-value: 0.054
#> H_0: no autocorrelation up to AR 1
#>
#> ------------------------------------------------------
#> Shapiro-Wilk Test for Normality
#> Test statistic: 0.965
#> p-value: 0.158
#> H_0: residuals are distributed normal
#>
#> ------------------------------------------------------
#> Log-likelihood: 148.094
#> AIC: -284.187
#> BIC: -272.96
#> Note: AIC and BIC calculated with k = 5 on T = 48 observations.
#>
#> ------------------------------------------------------
#> Breusch-Godfrey test indicates we reject the null hypothesis of no autocorrelation at p < 0.10.
#> Add lags to remove autocorrelation before running dynardl simulations.At a reasonable level of significance (\(p
< 0.10\)), the BG test indicates that we reject the null
hypothesis of no autocorrelation in the residuals of the model in
res. Philips (2018) indicates
that the next step is to add lagged first-differences to our model. To
add a lagged difference of \(y\), we
would run
res2 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = FALSE)
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
summary(res2)
#>
#> Call:
#> lm(formula = as.formula(paste(paste(dvnamelist), "~", paste(colnames(IVs),
#> collapse = "+"), collapse = " ")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.027161 -0.006046 0.001101 0.007727 0.026620
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.145837 0.031144 4.683 3.09e-05 ***
#> l.1.concern -0.195132 0.052971 -3.684 0.000665 ***
#> ld.1.concern -0.195748 0.131225 -1.492 0.143434
#> d.1.incshare10 0.679022 0.302936 2.241 0.030471 *
#> d.1.urate 0.001068 0.001665 0.641 0.524878
#> l.1.incshare10 -0.085775 0.032804 -2.615 0.012434 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.01144 on 41 degrees of freedom
#> (2 observations deleted due to missingness)
#> Multiple R-squared: 0.4174, Adjusted R-squared: 0.3464
#> F-statistic: 5.875 on 5 and 41 DF, p-value: 0.0003507and then again test for autocorrelation.
dynardl.auto.correlated(res2)
#>
#> ------------------------------------------------------
#> Breusch-Godfrey LM Test
#> Test statistic: 0.453
#> p-value: 0.501
#> H_0: no autocorrelation up to AR 1
#>
#> ------------------------------------------------------
#> Shapiro-Wilk Test for Normality
#> Test statistic: 0.986
#> p-value: 0.857
#> H_0: residuals are distributed normal
#>
#> ------------------------------------------------------
#> Log-likelihood: 146.636
#> AIC: -279.271
#> BIC: -266.32
#> Note: AIC and BIC calculated with k = 6 on T = 47 observations.
#>
#> ------------------------------------------------------
length(res2$model$residuals)
#> [1] 47We can now be much more confident that there is no more autocorrelation in the residuals. At this point, we can execute the ARDL-bounds test procedure. We need a few pieces of information: the length of the time series, the \(t\)-statistic on the lagged dependent variable in the ARDL error-correction model, the “case” of the regression (a combination of whether the intercept, if any, and trend, if any, were restricted (Pesaran, Shin, and Smith 2001)), and the number of regressors \(k\) appearing in first lags (NOT including the lagged dependent variable). We also need the number of observations in the model time series and the \(F\)-statistic on the restriction that the first lags of each of the variables in the model (including the lagged dependent variable) are jointly equal to zero.
From our regression, we know the \(t\)-statistic on the lagged dependent
variable is -3.684 (just looking at the output). Additionally, we
estimated a model with an unrestricted intercept and no trend, which
happens to be case 3 (which we would know by referencing Pesaran, Shin,
and Smith (2001)). There are \(k = 1\) variables in first lags
(incshare10, not including the LDV), and the number of
obs in the model is 47 periods. We now only need the test
of the restriction that the first lags are equal to zero. We can
calculate this by hand through coefficient testing. If we have all of
the coefficients, we just need to compare them to the set that are
lagged levels:
coef(res2$model)
#> (Intercept) l.1.concern ld.1.concern d.1.incshare10 d.1.urate
#> 0.145837457 -0.195131693 -0.195747775 0.679022362 0.001067503
#> l.1.incshare10
#> -0.085775490
# The second coefficient is the LDV, and the last coefficient is also a first lag. So they're both restricted
B <- coef(res2$model)
V <- vcov(res2$model)
# tag restrictions on LDV and l.1.incshare10
R <- matrix(c(0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1), byrow = T, nrow = 2)
k.plus1 <- sum(R)
# Restriction is that it is equal to 0
q <- 0
fstat <- (1/k.plus1)*t(R%*%B-q)%*%solve(R%*%V%*%t(R))%*%(R%*%B-q)
fstat
#> [,1]
#> [1,] 12.20351To test for cointegration, we need special critical values for the
F-test statistic. For this, we’re ready for pssbounds
pssbounds(obs = 47, fstat = 12.20351, tstat = -3.684, case = 3, k = 1)
#>
#> PESARAN, SHIN AND SMITH (2001) COINTEGRATION TEST
#>
#> Observations: 47
#> Number of Lagged Regressors (not including LDV) (k): 1
#> Case: 3 (Unrestricted intercept; no trend)
#>
#> ------------------------------------------------------
#> - F-test -
#> ------------------------------------------------------
#> <------- I(0) ------------ I(1) ----->
#> 10% critical value 4.19 4.94
#> 5% critical value 5.22 6.07
#> 1% critical value 7.56 8.69
#>
#>
#> F-statistic = 12.204
#> ------------------------------------------------------
#> - t-test -
#> ------------------------------------------------------
#> <------- I(0) ------------ I(1) ----->
#> 10% critical value -2.57 -2.91
#> 5% critical value -2.86 -3.22
#> 1% critical value -3.43 -3.82
#>
#>
#> t statistic = -3.684
#> ------------------------------------------------------
#>
#> t-statistic note: Small-sample critical values not provided for Case III. Asymptotic critical values used.Finally, a payoff. The \(t\)-statistic and \(F\)-statistic are situated between special sample critical values. Depending on the level of significance that we preselected, these values offer a number of different conclusions. If the \(F\)-statistic exceeds the upper I(1) critical value, we may conclude cointegration. Thus, we can be confident that we have a well-specified model. If the F-statistic falls below the I(0) critical value, Pesaran, Shin, and Smith (2001) note that this indicates that all regressors are in fact stationary. Thus, cointegration cannot exist. If the F-statistic falls between the lower I(0) and upper I(1) critical values, the results are inconclusive. This means that cointegration may exist, but further testing and re-specification of the model is needed. Users that end up with these results are encouraged to see Philips (2018), who provides a strategy for dealing with this.
In our model here, since the value of the \(F\)-statistic exceeds the critical value at the upper I(1) bound of the test at the 5% level, we may conclude that Income Top 10 (the variable in the model in levels) and the dependent variable, Concern, are in a cointegrating relationship. This is furthered by the \(t\)-statistic of -3.684 falling below the 5% critical value for I(1). Thus, we can be confident both in our model specification and the unique, long-run relationship that exists between the two variables.
Calculating all of those values by hand was unnecessarily involved,
especially since we know most of the values are saved post-estimation
anyway. We’re further motivated not to have to reference the tables in
Pesaran, Shin, and Smith (2001) for the
case of the regression every time we run pssbounds. Thus,
our preferred way of estimating the ARDL-bounds test is just by passing
the error-correction model to pssbounds. In other words,
the test is equivalent by just running
pssbounds(res2)
#>
#> PESARAN, SHIN AND SMITH (2001) COINTEGRATION TEST
#>
#> Observations: 47
#> Number of Lagged Regressors (not including LDV) (k): 1
#> Case: 3 (Unrestricted intercept; no trend)
#>
#> ------------------------------------------------------
#> - F-test -
#> ------------------------------------------------------
#> <------- I(0) ------------ I(1) ----->
#> 10% critical value 4.19 4.94
#> 5% critical value 5.22 6.07
#> 1% critical value 7.56 8.69
#>
#>
#> F-statistic = 12.204
#> ------------------------------------------------------
#> - t-test -
#> ------------------------------------------------------
#> <------- I(0) ------------ I(1) ----->
#> 10% critical value -2.57 -2.91
#> 5% critical value -2.86 -3.22
#> 1% critical value -3.43 -3.82
#>
#>
#> t statistic = -3.684
#> ------------------------------------------------------
#>
#> t-statistic note: Small-sample critical values not provided for Case III. Asymptotic critical values used.Pesaran, Shin, and Smith (2001) also
provide for testing that the intercept is also restricted, which they
refer to as case 2. If we wanted to add this restriction to the test, we
would use restriction = TRUE in pssbounds.
pssbounds(res2, restriction = TRUE)
#>
#> PESARAN, SHIN AND SMITH (2001) COINTEGRATION TEST
#>
#> Observations: 47
#> Number of Lagged Regressors (not including LDV) (k): 1
#> Case: 2 (Intercept included in F-stat restriction; no trend)
#>
#> ------------------------------------------------------
#> - F-test -
#> ------------------------------------------------------
#> <------- I(0) ------------ I(1) ----->
#> 10% critical value 3.18 3.65
#> 5% critical value 3.86 4.44
#> 1% critical value 5.50 6.24
#>
#>
#> F-statistic = 8.137
#>
#> ------------------------------------------------------
#>
#> t-statistic note: Critical values do not currently exist for Case II.Of course, users are encouraged to read Pesaran, Shin, and Smith
(2001) and Philips (2018) for the implications of this test. We
merely note that cases 2 and 4 accessible through
restriction = TRUE. We also note that, for most users most
of the time, case 3 will be the most common case (an unrestricted
intercept and no trend).
Any dynardl object run in an error-correction format is
available for pssbounds post-estimation. This, of course,
is not meant to imply that all dynardl models are meant to
be tested for cointegration. Stationary dependent variables cannot be
cointegrated with other variables. As such, we would want to run those
models in levels, with EC = FALSE. Like always, we point
users to the importance of pre-regression testing of the nature of our
variables, most specifically whether or not they are integrated.
dynardl will attempt to be as helpful as possible, but in
the end, it is up to the user to know which model is appropriate and to
ensure that the model is specified correctly.
We had another motivation. dynardl +
pssbounds are a powerful combination for estimating and
testing for cointegration in a unified framework. But once we have
tested for cointegration, we still want inferences about the nature of
the effect of some \(x\) variable on
the dependent variable. And these inferences become much more difficult
to obtain as our models increase in complexity. For instance, the very
lagged difference of \(y\) that we
introduced to help purge autocorrelation from the model we just
estimated made interpreting the coefficients from that model much less
straightforward. In the next section, we elaborate on the ability of
dynardl to provide these inferences through the process we
outlined earlier: counterfactual simulation of a response to a shock in
some \(x\) variable. These inferences
do not require anything beyond what we have already covered: a
dynardl model and a theoretically interesting \(x\) variable.
Counterfactual simulation using dynardl
ARDL models are useful because they are flexible, but their flexibility undermines our ability to make sense of the coefficients estimated in any given model. An alternative approach to interpretation is to use the coefficients from an estimated model to simulate meaningful responses in the dependent variable to counterfactual changes in an independent variable \(x\) (that we control), allowing the change to filter through the various forms of the \(x\) variable in the model, as well as different forms of the \(y\) variable (like differences and lagged levels) that might be included.
dynamac handles this simulated interpretation through a function we
have already seen: dynardl. All we need to do is specify
that simulate = TRUE, and we will produce simulated
responses: we can observe how a change to a variable in a
dynardl model produces a corresponding change in the
dependent variable. Other arguments are required, but only one has no
default: we need to know which \(x\)
variable to simulate a shock to (the shockvar). This
“shock” means that, at a time specified by the user, the value of an
\(x\) variable will move to some level.
If the variable is in levels or lagged levels, this means that its new
value becomes the pre-shock average plus whatever the shock value is. If
the variable is in differences or lagged differences, the shock lasts
for one period (as a permanent change in a differenced variable would
imply that it is changing every period!).
dynardl has reasonable defaults for all of the other
required parameters: simulations default to a range of 20
periods, lines are drawn at roughly sig = 95% confidence,
the shockvar is shocked by a standard deviation (the
shockval), we use 1,000 sims to simulate the
average response, we don’t force the other \(x\) variables to any values (we allow them
to take their means, except for differenced variables, which we set to
be zero: assuming that, period-over-period, there is no change), we
allow for 20 periods of burnin, and we create predicted
values of the dependent variable, rather than expected values. All of
these options can be controlled by the user, but we’ll return to that in
a moment.
The simulation process is fairly straightforward.
dynardl uses the coefficients from the model. It draws a
number (specifically, sims) of values of the coefficients
\(\hat\beta\) from the estimated
regression model. The distribution is assumed to be multivariate normal
with mean of \(\hat\beta\) and
variance-covariance of the estimated model. Uncertainty is incorporated
by simulating \(\hat\sigma\) \(^{2}\) as a scaled draw from the
chi-squared distribution. These fit together by using values of the
independent variables (usually their means: see the preceding paragraph)
\(X\), multiplying by \(\hat\beta\) to get predicted values of the
dependent variable \(y\), and then
using \(\hat \sigma\) \(^{2}\) to introduce uncertainty in the
predicted values. If you want to exert more direct control over the
values of any \(x\) variables used, you
can forceset them to any other value you like. This is not
limited to means or integers; if you have any substantively interesting
value you’d like to hold a variable at, you are free to specify
whichever value you like.
But what we’re really interested in is the effect of some variable
\(x\). In the world of
dynardl, this is our shockvar. We specify a
variable from the model, tell dynardl how much we want it
to theoretically change by (in shockval, defaulting to a
standard deviation of the shockvar) and when we want it to
change (time), and then observe the change in \(y\).
Let’s go back to our earlier model. Remember we ran
res2 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = FALSE)
summary(res2)Here, we set simulate = FALSE because we were just
illustrating estimation without simulations (given that simulations take
a few seconds to estimate, we may only want to simulate changes when we
have a final model, free of autocorrelation and the other things we
tested for). Now, we’ll turn simulate = TRUE and specify an
\(x\) variable to observe the response
of. We’ll observe the changes resulting from incshare10,
given its lagged level demonstrated a significant effect. In other
words, our shockvar = "incshare10". By default,
dynardl will shock it by its standard deviation, but we can
observe any change we like with shockval. As with any other
time in which stochastic values are involved, we should set a seed to
ensure our results are directly replicable.
set.seed(020990)
res2 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = TRUE,
shockvar = "incshare10")
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
#> [1] "incshare10 shocked by one standard deviation of incshare10 by default."
#> [1] "dynardl estimating ..."
#>
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summary(res2$model)
#>
#> Call:
#> lm(formula = as.formula(paste(paste(dvnamelist), "~", paste(colnames(IVs),
#> collapse = "+"), collapse = " ")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.027161 -0.006046 0.001101 0.007727 0.026620
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.145837 0.031144 4.683 3.09e-05 ***
#> l.1.concern -0.195132 0.052971 -3.684 0.000665 ***
#> ld.1.concern -0.195748 0.131225 -1.492 0.143434
#> d.1.incshare10 0.679022 0.302936 2.241 0.030471 *
#> l.1.incshare10 -0.085775 0.032804 -2.615 0.012434 *
#> d.1.urate 0.001068 0.001665 0.641 0.524878
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.01144 on 41 degrees of freedom
#> (2 observations deleted due to missingness)
#> Multiple R-squared: 0.4174, Adjusted R-squared: 0.3464
#> F-statistic: 5.875 on 5 and 41 DF, p-value: 0.0003507It looks somewhat goofy in the vignette, as dynardl
displays a progress bar to inform you of how many simulations have been
completed. But the payoff is in res2$simulation. We get to
observe the effect of a standard-deviation shock in
incshare10 in the dependent variable. This response is best
visualized in a plot. To see this plot, the model can be saved to an
object, and plots can be created with
dynardl.simulation.plot.
dynardl.simulation.plot(res2, type = "area", response = "levels")
Here, the shock has an immediate effect that does not dissipate for a
long time. So long, in fact, that we might want to lengthen the
range of the simulation.
set.seed(020990)
res3 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = TRUE, range = 30,
shockvar = "incshare10")
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
#> [1] "incshare10 shocked by one standard deviation of incshare10 by default."
#> [1] "dynardl estimating ..."
#>
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dynardl.simulation.plot(res3, type = "area", response = "levels")
If instead of an area plot we desired a spike plot:
dynardl.simulation.plot(res3, type = "spike", response = "levels")
In res3 there are actually three sets of output. The
first is the model, the second is the information for
pssbounds, and the last is for plotting. We mention this so
that users can use their usual TeX shortcuts for creating tables,
customizing plots, or testing using pssbounds later.
res3$model
#>
#> Call:
#> lm(formula = as.formula(paste(paste(dvnamelist), "~", paste(colnames(IVs),
#> collapse = "+"), collapse = " ")))
#>
#> Coefficients:
#> (Intercept) l.1.concern ld.1.concern d.1.incshare10 l.1.incshare10
#> 0.145837 -0.195132 -0.195748 0.679022 -0.085775
#> d.1.urate
#> 0.001068
res3$pssbounds
#> NULL
res3$simulation
#> time central ll95 ll90 ll75 ul75 ul90 ul95
#> 1 1 0.5792020 0.5559682 0.5592160 0.5651339 0.5925600 0.5991881 0.6018618
#> 2 2 0.5787013 0.5554403 0.5600252 0.5654825 0.5917471 0.5974948 0.6014550
#> 3 3 0.5789356 0.5570463 0.5608863 0.5668301 0.5918044 0.5975220 0.6000707
#> 4 4 0.5787997 0.5551703 0.5592707 0.5650761 0.5922909 0.5979111 0.6012698
#> 5 5 0.5793091 0.5542530 0.5602907 0.5658583 0.5931276 0.5988996 0.6030645
#> 6 6 0.5795942 0.5571192 0.5600127 0.5667194 0.5924350 0.5988118 0.6033832
#> 7 7 0.5798832 0.5581873 0.5605118 0.5657941 0.5944158 0.5999712 0.6036243
#> 8 8 0.5802234 0.5593644 0.5619827 0.5672503 0.5935926 0.5994727 0.6040383
#> 9 9 0.5799968 0.5577335 0.5610041 0.5665941 0.5933536 0.6005407 0.6045266
#> 10 10 0.6167511 0.5797056 0.5859896 0.5951996 0.6386637 0.6465144 0.6532313
#> 11 11 0.5973816 0.5713986 0.5756398 0.5816251 0.6119134 0.6175554 0.6226215
#> 12 12 0.5928346 0.5692578 0.5727756 0.5786376 0.6069353 0.6135546 0.6161177
#> 13 13 0.5856054 0.5618984 0.5655589 0.5715046 0.5995451 0.6052888 0.6104257
#> 14 14 0.5809483 0.5572153 0.5613369 0.5670413 0.5950571 0.6012514 0.6051003
#> 15 15 0.5763318 0.5541748 0.5574485 0.5625207 0.5899999 0.5960573 0.5998357
#> 16 16 0.5727459 0.5492940 0.5532698 0.5591775 0.5864000 0.5930059 0.5968420
#> 17 17 0.5703495 0.5464895 0.5504217 0.5564761 0.5840667 0.5899451 0.5941536
#> 18 18 0.5672469 0.5449277 0.5480448 0.5539188 0.5807144 0.5869591 0.5911982
#> 19 19 0.5653370 0.5419017 0.5455841 0.5514246 0.5786327 0.5847831 0.5902170
#> 20 20 0.5636972 0.5388558 0.5430262 0.5495522 0.5778304 0.5833093 0.5869392
#> 21 21 0.5625315 0.5387461 0.5422292 0.5486370 0.5763966 0.5820638 0.5874475
#> 22 22 0.5615651 0.5397573 0.5425750 0.5477181 0.5743292 0.5819248 0.5855887
#> 23 23 0.5602949 0.5357544 0.5400916 0.5464939 0.5742114 0.5800424 0.5847322
#> 24 24 0.5592019 0.5368418 0.5398974 0.5463206 0.5729166 0.5789579 0.5824619
#> 25 25 0.5584347 0.5346550 0.5389714 0.5445627 0.5725257 0.5790650 0.5822461
#> 26 26 0.5578433 0.5336365 0.5377534 0.5429992 0.5720966 0.5782946 0.5832261
#> 27 27 0.5579985 0.5348062 0.5376151 0.5437441 0.5717968 0.5774593 0.5817657
#> 28 28 0.5566465 0.5326561 0.5369266 0.5424172 0.5703003 0.5769466 0.5808367
#> 29 29 0.5569417 0.5340579 0.5369856 0.5440453 0.5700653 0.5755345 0.5788088
#> 30 30 0.5565073 0.5328118 0.5361860 0.5422267 0.5701398 0.5762442 0.5801271
#> d.central d.ll95 d.ll90 d.ll75 d.ul75
#> 1 0.0004254538 -0.0228083716 -0.019560503 -0.01364261 0.013783426
#> 2 -0.0005007145 -0.0237617318 -0.019176795 -0.01371949 0.012545130
#> 3 0.0002343734 -0.0216549368 -0.017814920 -0.01187115 0.013103084
#> 4 -0.0001358976 -0.0237653150 -0.019664922 -0.01385957 0.013355292
#> 5 0.0005093204 -0.0245467083 -0.018509030 -0.01294141 0.014327884
#> 6 0.0002850980 -0.0221898171 -0.019296377 -0.01258964 0.013125935
#> 7 0.0002889919 -0.0214068575 -0.019082324 -0.01380002 0.014821609
#> 8 0.0003402761 -0.0205187529 -0.017900491 -0.01263282 0.013709454
#> 9 -0.0002265877 -0.0224899460 -0.019219351 -0.01362931 0.013130146
#> 10 0.0367542274 -0.0002912449 0.005992723 0.01520280 0.058666890
#> 11 -0.0193695051 -0.0453524455 -0.041111309 -0.03512593 -0.004837701
#> 12 -0.0045469415 -0.0281237411 -0.024605934 -0.01874393 0.009553755
#> 13 -0.0072292034 -0.0309362178 -0.027275693 -0.02133002 0.006710462
#> 14 -0.0046571172 -0.0283901378 -0.024268563 -0.01856416 0.009451657
#> 15 -0.0046164894 -0.0267735305 -0.023499795 -0.01842756 0.009051613
#> 16 -0.0035858975 -0.0270377883 -0.023061974 -0.01715427 0.010068223
#> 17 -0.0023964171 -0.0262563905 -0.022324247 -0.01626981 0.011320804
#> 18 -0.0031025991 -0.0254217833 -0.022304670 -0.01643069 0.010364867
#> 19 -0.0019098826 -0.0253452050 -0.021662811 -0.01582229 0.011385780
#> 20 -0.0016398588 -0.0264812467 -0.022310866 -0.01578478 0.012493344
#> 21 -0.0011656724 -0.0249511070 -0.021467954 -0.01506013 0.012699410
#> 22 -0.0009664072 -0.0227742211 -0.019956522 -0.01481340 0.011797668
#> 23 -0.0012701799 -0.0258106578 -0.021473496 -0.01507121 0.012646361
#> 24 -0.0010929582 -0.0234530774 -0.020397472 -0.01397435 0.012621721
#> 25 -0.0007672501 -0.0245469469 -0.020230557 -0.01463928 0.013323770
#> 26 -0.0005913811 -0.0247981816 -0.020681296 -0.01543551 0.013661891
#> 27 0.0001551995 -0.0230371348 -0.020228196 -0.01409920 0.013953479
#> 28 -0.0013519643 -0.0253424119 -0.021071924 -0.01558132 0.012301787
#> 29 0.0002951263 -0.0225886396 -0.019660904 -0.01260129 0.013418715
#> 30 -0.0004343467 -0.0241298890 -0.020755687 -0.01471502 0.013198078
#> d.ul90 d.ul95 shocktime
#> 1 0.020411585 0.023085233 10
#> 2 0.018292793 0.022253016 10
#> 3 0.018820762 0.021369451 10
#> 4 0.018975487 0.022334112 10
#> 5 0.020099900 0.024264724 10
#> 6 0.019502703 0.024074178 10
#> 7 0.020376991 0.024030172 10
#> 8 0.019589539 0.024155147 10
#> 9 0.020317277 0.024303148 10
#> 10 0.066517512 0.073234459 10
#> 11 0.000804376 0.005870427 10
#> 12 0.016173082 0.018736182 10
#> 13 0.012454184 0.017591054 10
#> 14 0.015646006 0.019494915 10
#> 15 0.015108969 0.018887356 10
#> 16 0.016674052 0.020510154 10
#> 17 0.017199230 0.021407646 10
#> 18 0.016609624 0.020848743 10
#> 19 0.017536248 0.022970073 10
#> 20 0.017972330 0.021602221 10
#> 21 0.018366642 0.023750308 10
#> 22 0.019393270 0.023057248 10
#> 23 0.018477312 0.023167130 10
#> 24 0.018663032 0.022166954 10
#> 25 0.019863015 0.023044154 10
#> 26 0.019859865 0.024791362 10
#> 27 0.019616014 0.023922348 10
#> 28 0.018948054 0.022838160 10
#> 29 0.018887972 0.022162296 10
#> 30 0.019302571 0.023185472 10Only models where ec = TRUE will have
$pssbounds information, as only those models can possibly
be testing a cointegrating relationship. We suspect that no one will
need them by hand, as you can just pass the whole object via
pssbounds(res3) to get the same results.
Other options might be of interest. In order to smooth our confidence
intervals (note: this does NOT make us more confident), we can increase
the number of simulations. Additionally, the plotting function allows
the user to produce plot in grayscale (for publications). Consider the
following (notice the sims argument of dynardl
and the new arguments under dynardl.simulation.plot:
set.seed(020990)
res4 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = TRUE, range = 30, sims = 10000,
shockvar = "incshare10")
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
#> [1] "incshare10 shocked by one standard deviation of incshare10 by default."
#> [1] "dynardl estimating ..."
#>
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dynardl.simulation.plot(res4, type = "spike", response = "levels", bw = TRUE)
The full extent of these options is addressed in the documentation.
There is some question as to whether or not quantities of interest
from these types of stochastic simulations are best summarized by the
means or the medians of the resulting distributions. In most cases, the
difference is likely to be minimal. In cases of skew, though, the median
might serve significantly better than the mean (described in Rainey
(2017)). Here, we can do that by setting
qoi = median.
set.seed(020990)
res5 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = TRUE, range = 30,
shockvar = "incshare10", qoi = "median")
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
#> [1] "incshare10 shocked by one standard deviation of incshare10 by default."
#> [1] "dynardl estimating ..."
#>
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dynardl.simulation.plot(res5, type = "area", response = "levels")
There are a variety of quantities that are plottable from the
simulations, outside of just the response of the dependent variable.
dynardl.simulation.plot includes options for plotting the
levels of the dependent variable, the levels but demeaned
(levels.from.mean) of the dependent variable, and the
period-over-period diffs of the changes in the simulated
values. You can get a sense of how the shock to the independent variable
is decaying through time by observing the differences in each period as
an absolute value (how much is the dependent variable adjusting) through
shock.effect.decay. You can also see the
cumulative.diffs of those changes and the absolute value of
the changes cumulative.abs.diffs. For the final two
options, fullsims = TRUE must be specified in the
dynardl simulation, which reports the full raw simulated
values as a part of the dynardl object. These values are
used to draw confidence intervals over the simulated changes.
In addition, dynardl.simulation.plot will expect a value
for when it should regard the changes in the dependent variable as
noise, rather than real changes. The default tol
is to disregard changes that are less than 0.1% of the mean of the
dependent variable. Alternatively, we can specify a
last.period in which we believe the dependent variable is
responding. dynardl.simulation.plot also allows you to pass
whatever normal plotting arguments you would use in the normal
... way.
set.seed(020990)
res6 <- dynardl(concern ~ incshare10 + urate, data = ineq,
lags = list("concern" = 1, "incshare10" = 1),
diffs = c("incshare10", "urate"),
lagdiffs = list("concern" = 1),
ec = TRUE, simulate = TRUE, range = 30,
shockvar = "incshare10", fullsims = TRUE)
#> [1] "Error correction (EC) specified; dependent variable to be run in differences."
#> [1] "incshare10 shocked by one standard deviation of incshare10 by default."
#> [1] "dynardl estimating ..."
#>
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par(mfrow = c(2, 3))
dynardl.simulation.plot(res6, type = "area", response = "levels")
dynardl.simulation.plot(res6, type = "area", response = "levels.from.mean")
dynardl.simulation.plot(res6, type = "area", response = "diffs")
dynardl.simulation.plot(res6, type = "area", response = "shock.effect.decay")
dynardl.simulation.plot(res6, type = "area", response = "cumulative.diffs", axes = F)
dynardl.simulation.plot(res6, type = "area", response = "cumulative.abs.diffs")
#> Warning in dynardl.simulation.plot(res6, type = "area", response =
#> "cumulative.abs.diffs"): Cumulative absolute effects assumed to be noise (by
#> tolerance) at t = 13.
If we don’t want to see the equilibriating behavior before the shock,
we can draw the plot starting in a different period through
start.period.
dynardl.simulation.plot(res6, response = "shock.effect.decay", start.period = 9)
If you’re in an exploratory phase of model building and prefer not to
have to run the plots separately, you are free to combine all of them
using dynardl.all.plots.
dynardl.all.plots(res6)
#> Warning in dynardl.simulation.plot(x, response = "cumulative.abs.diffs", :
#> Cumulative absolute effects assumed to be noise (by tolerance) at t = 13.