The PSTR package implements the Panel Smooth Transition Regression (PSTR) modelling. You can find the package on CRAN, see
The modelling procedure consists of three stages: Specification, Estimation and Evaluation. The package offers tools helping the users to conduct model specification tests, to do PSTR model estimation, and to do model evaluation.
The cluster-dependency and heteroskedasticity-consistent tests are implemented in the package.
The wild bootstrap and cluster wild bootstrap tests are also implemented.
Parallel computation (as an option) is implemented in some functions, especially the bootstrap tests. Therefore, the package suits tasks running many cores on super-computation servers.
You can install the package by running
devtools::install_github("yukai-yang/PSTR")or You can install the package by running
remotes::install_github("yukai-yang/PSTR")provided that the package devtools or
remotes has been installed beforehand.
After installing the package, you need to load (attach better say) it by running the code
library(PSTR)You can first check the information and the current version number by running
version()
#> PSTR version 2.0.0 (Green Panel)Then you can take a look at all the available functions and data in the package
ls( grep("PSTR", search()) )
#> [1] "EstPSTR" "EvalTest" "Hansen99"
#> [4] "LinTest" "NewPSTR" "plot_coefficients"
#> [7] "plot_response" "plot_target" "plot_transition"
#> [10] "sunspot" "version" "WCB_HETest"
#> [13] "WCB_LinTest" "WCB_TVTest"In the package, a data set called Hansen99 is offered to
give prompt example. For details of the data set, you can run
?Hansen99 You can create a new object of the class PSTR by doing
pstr = NewPSTR(Hansen99, dep='inva', indep=4:20, indep_k=c('vala','debta','cfa','sales'),
tvars=c('vala'), im=1, iT=14)
#> ✔ The PSTR model is ready.
pstr
#>
#> ── R package PSTR 2.0.0 (Green Panel) ──────────────────────────────────────────
#>
#> ── Summary of the model ──
#>
#> The long format panel is 14 x 560 (time x individual).
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> ── Dependent variable
#> inva
#>
#> ── Explanatory variables in the linear part (17)
#> dt_75, dt_76, dt_77, dt_78, dt_79, dt_80, dt_81, dt_82, dt_83, dt_84, dt_85,
#> dt_86, dt_87, vala, debta, cfa, sales
#>
#> ── Explanatory variables in the non-linear part (4)
#> vala, debta, cfa, sales
#>
#> ── Potential transition variable(s) to be tested (1)
#> valaIt says that the data set Hansen99 is used, the
dependent variable is inva, the variables in the data from
column 4 to 20 are the explanatory variables in the linear part (though
you can write down the names of them), the explanatory variables in the
nonlinear part are the four ones in indep_k, and the
potential transition variable is vala (Tobin’s Q).
Now you can see that the NewPSTR is basically defining
the settings of the model.
Note that you can print the object of the class PSTR. By
default, it gives you a summary of the PSTR model. They are mainly about
which one is the dependent variable, which ones are explanatory
variables and etc..
The following code does linearity tests
LinTest(pstr)
#> ✔ Done!
print(pstr, mode="tests")
#>
#> ── R package PSTR 2.0.0 (Green Panel) ──────────────────────────────────────────
#>
#> ── Results of the linearity (homogeneity) tests ──
#>
#> ── LM tests based on transition variable 'vala'
#>
#>
#> m LM_X LM_F HAC_X HAC_F
#> ------ --------- -------- ------- -------
#> 1 125.2509 28.9922 30.033 6.9518
#> p-val 0.0000 0.0000 0.000 0.0000
#>
#> ── Sequence of homogeneity tests for selecting number of switches 'm' ──
#>
#> ── LM tests based on transition variable 'vala'
#>
#>
#> m LM_X LM_F HAC_X HAC_F
#> ------ --------- -------- ------- -------
#> 1 125.2509 28.9922 30.033 6.9518
#> p-val 0.0000 0.0000 0.000 0.0000or
pstr$LinTest()
print(pstr, mode="tests")LinTest(pstr) modifies the object in place (R6 uses
reference semantics). The function returns pstr invisibly,
so you may call it either for its side effects or in an assignment. The
reference semantics means that the pass of the object to the function
leads to a change of the values inside the object.
You can create a new PSTR objects (the same model with the same data set) based on another object by running
pstr0 = pstr$clone()By doing so, the new PSTR object pstr0 is an independent
copy of the original object, and when you do something on either of the
two objects (pstr or pstr0), the other one
remains unchanged.
You can do, for example, the wild bootstrap and wild cluster bootstrap by running the following code.
iB = 5000 # the number of repetitions in the bootstrap
library(snowfall)
WCB_LinTest(pstr,iB=iB,parallel=T,cpus=50)It takes a long long time to run the bootstrap. This function is
developed for those who work on some super-computation server with many
cores and a large memory. Note that you will have to attach the
snowfall package manually.
You can try the following code on your own computer by reducing the number of repetitions and cores.
WCB_LinTest(pstr,iB=4,parallel=T,cpus=2)After selecting the transition variable (in this example,
vala), you can estimate the PSTR model.
EstPSTR(use=pstr,im=1,iq=1,useDelta=T,par=c(-0.462,0), vLower=4, vUpper=4)
print(pstr, mode="estimates")By default, the optim method L-BFGS-B is
used, but you can change the method for estimation by doing
EstPSTR(use=pstr,im=1,iq=1,useDelta=T,par=c(-0.462,0), method="CG")
print(pstr, mode="estimates")
#>
#> ── R package PSTR 2.0.0 (Green Panel) ──────────────────────────────────────────
#>
#> ── Results of the PSTR estimation: ──
#>
#> ℹ Transition variable 'vala' is used in the estimation.
#>
#> ── Parameter estimates in the linear part (first extreme regime)
#>
#>
#> dt_75_0 dt_76_0 dt_77_0 dt_78_0 dt_79_0
#> -------- ---------- ---------- ---------- ---------- ---------
#> Est -0.002827 -0.007512 -0.005812 0.0003951 0.002464
#> s.e. 0.002431 0.002577 0.002649 0.0027950 0.002708
#> t-ratio -1.163000 -2.915000 -2.193000 0.1414000 0.909700
#>
#>
#>
#> dt_80_0 dt_81_0 dt_82_0 dt_83_0 dt_84_0
#> -------- --------- ---------- ---------- ---------- -----------
#> Est 0.006085 0.0004164 -0.007802 -0.014410 -0.0009146
#> s.e. 0.002910 0.0029220 0.002609 0.002701 0.0030910
#> t-ratio 2.091000 0.1425000 -2.990000 -5.337000 -0.2959000
#>
#>
#>
#> dt_85_0 dt_86_0 dt_87_0 vala_0 debta_0
#> -------- --------- ---------- ---------- -------- ---------
#> Est 0.003467 -0.001591 -0.008606 0.11500 -0.03392
#> s.e. 0.003232 0.003202 0.003133 0.04073 0.03319
#> t-ratio 1.073000 -0.496900 -2.747000 2.82200 -1.02200
#>
#>
#>
#> cfa_0 sales_0
#> -------- -------- ---------
#> Est 0.10980 0.002978
#> s.e. 0.04458 0.008221
#> t-ratio 2.46400 0.362200
#>
#> ── Parameter estimates in the non-linear part
#>
#>
#> vala_1 debta_1 cfa_1 sales_1
#> -------- --------- -------- --------- ---------
#> Est -0.10370 0.02892 -0.08801 0.005945
#> s.e. 0.03981 0.04891 0.05672 0.012140
#> t-ratio -2.60300 0.59140 -1.55200 0.489600
#>
#> ── Parameter estimates in the second extreme regime
#>
#>
#> vala_{0+1} debta_{0+1} cfa_{0+1} sales_{0+1}
#> -------- ----------- ------------ ---------- ------------
#> Est 0.011300 -0.00500 0.02183 0.008923
#> s.e. 0.001976 0.01739 0.01885 0.004957
#> t-ratio 5.718000 -0.28750 1.15800 1.800000
#>
#> ── Non-linear parameter estimates
#>
#>
#> gamma c_1
#> -------- ------- -----------
#> Est 0.6299 -0.0002008
#> s.e. 0.1032 0.7252000
#> t-ratio 6.1040 -0.0002769
#> ℹ Estimated standard deviation of the residuals is 0.04301.
#> ℹ The specification results are ready, run `print(obj, mode="tests")` to show
#> the results.The argument useDelta determines the type of the initial
value for the smoothness parameter. By default
useDelta = FALSE means that the first initial value in
par is the gamma instead of
delta. Here we use the settings
useDelta = TRUE and par = c(1.6, .5) means
that the first value of par is the delta and
its value is 1.6. Note that delta and gamma
have the relationship gamma = exp(delta). Thus, the
following two sentences are equivalent
EstPSTR(use=pstr,im=1,iq=1,useDelta=T,par=c(-0.462,0), method="CG")
EstPSTR(use=pstr,im=1,iq=1,par=c(exp(-0.462),0), method="CG")Note that the estimation of a linear panel regression model is also implemented. The user can do it by simply running
EstPSTR(use=pstr0)
print(pstr0, mode="estimates")
#>
#> ── R package PSTR 2.0.0 (Green Panel) ──────────────────────────────────────────
#>
#> ── A linear panel regression with fixed effects is estimated. ──
#>
#> ── Parameter estimates
#>
#>
#> dt_75 dt_76 dt_77 dt_78 dt_79
#> -------- ---------- ---------- ---------- --------- ---------
#> Est -0.007759 -0.008248 -0.004296 0.002356 0.004370
#> s.e. 0.002306 0.002544 0.002718 0.002820 0.002753
#> t-ratio -3.365000 -3.242000 -1.581000 0.835600 1.587000
#>
#>
#>
#> dt_80 dt_81 dt_82 dt_83 dt_84
#> -------- --------- --------- ---------- ---------- ---------
#> Est 0.008246 0.004164 -0.005294 -0.010040 0.006864
#> s.e. 0.002959 0.002992 0.002664 0.002678 0.003092
#> t-ratio 2.786000 1.391000 -1.987000 -3.749000 2.220000
#>
#>
#>
#> dt_85 dt_86 dt_87 vala debta
#> -------- --------- --------- ---------- --------- ----------
#> Est 0.009740 0.007027 0.0004091 0.008334 -0.016380
#> s.e. 0.003207 0.003069 0.0030080 0.001259 0.005725
#> t-ratio 3.037000 2.290000 0.1360000 6.619000 -2.861000
#>
#>
#>
#> cfa sales
#> -------- -------- ---------
#> Est 0.06506 0.007957
#> s.e. 0.01079 0.002412
#> t-ratio 6.02800 3.299000
#> ℹ Estimated standard deviation of the residuals is 0.04375.
#> ℹ The specification results are ready, run `print(obj, mode="tests")` to show
#> the results.The evaluation tests can be conducted after estimating a nonlinear PSTR model.
## evaluation tests
EvalTest(use=pstr,vq=as.matrix(Hansen99[,'vala'])[,1])Note that EvalTest takes one transition variable
vq at a time for the no remaining nonlinearity
(heterogeneity) test. This differs from LinTest, which can
evaluate several candidate transition variables through
tvars.
The user can also run the wild bootstrap (WB) and wild cluster bootstrap (WCB) versions of the evaluation tests, provided that suitable computing resources are available.
iB = 5000
cpus = 50
## wild bootstrap time-varying evaluation test
WCB_TVTest(use=pstr,iB=iB,parallel=T,cpus=cpus)
## wild bootstrap heterogeneity evaluation test
WCB_HETest(use=pstr,vq=as.matrix(Hansen99[,'vala'])[,1],iB=iB,parallel=T,cpus=cpus)
print(pstr, mode="evaluation")Note that the evaluation functions do not accept the returned object
pstr0 from a linear panel regression model, as the
evaluation tests are designed for the estimated PSTR model but not a
linear one.
After estimating the PSTR model, you can plot the estimated transition function by running
plot_transition(pstr)
or a better plot with more arguments
plot_transition(pstr, fill='blue', xlim=c(-2,20), color = "dodgerblue4", size = 2, alpha=.3) +
ggplot2::geom_vline(ggplot2::aes(xintercept = pstr$c - log(1/0.95 - 1)/pstr$gamma),color='blue') +
ggplot2::labs(x="customize the label for x axis",y="customize the label for y axis",
title="The Title",subtitle="The subtitle",caption="Make a caption here.")
You can also plot the curves of the coefficients, the standard errors and the p-values against the transition variable.
ret = plot_coefficients(pstr, vars=1:4, length.out=100, color="dodgerblue4", size=2)
ret[[1]]
The plotting function plot_response, which depicts the
relationship between which we refer to as the response, as a function of
an explanatory variable xit and the
transition variable qit.
The response [ϕ0 + ϕ1git(qit; γ, c)]xit is actually the contribution of the variable xit to the conditional expectation of the dependent variable y*i**t* through the smooth transition mechanism.
We can see that the response against the variable is a straight line if there is no nonlinearity. If xit and qit are distinct, the response can be visualised as a surface over (xit, qit), where the vertical axis represents the response. And it becomes a curve if the variable xit and the transition variable qit are identical.
You can generate these plots by running
ret = plot_response(obj=pstr, vars=1:4, log_scale = c(F,T), length.out=100)ret takes the return value of the function. We make the
graphs for all the four variables in nonlinear part by using
vars=1:4 (variable names can also be used for
specification). Note that we do not do it for the variables in the
linear part, as they produce straight lines or planes.
log_scale is a length-2 logical vector indicating whether
to apply a log scale to (i) xit and
(ii) qit, respectively.
length.out gives the number of points in the grid for
producing the surface or curve. A length.out of 100 points
looks fine enough.
You might ask: what if you want different log-scaling choices for different variables? The solution is to make the graphs separately by running something like
ret1 = plot_response(obj=pstr, vars=1, log_scale = c(F,T), length.out=100)
ret2 = plot_response(obj=pstr, vars=2, log_scale = c(T,T), length.out=100)Let us take a look at the elements in ret
attributes(ret)
#> $names
#> [1] "vala" "debta" "cfa" "sales"We see that ret is a list containing elements whose
names are the variables’ names that we specified when running
plot_response.
Yes, but each element is a ggplot object, so you can display it directly, for example:
ret$vala
The numbers on the x-axis look not so good as it is difficult to find where the turning-point is.
The ggplot2 package allows us to manually paint the
numbers (the PSTR package collaborates very well with some prevailing
packages), and even the label on x-axis (and many more).
ret$vala + ggplot2::scale_x_log10(breaks=c(.02,.05,.1,.2,.5,1,2,5,10,20)) +
ggplot2::labs(x="Tobin's Q")
Now we see very clearly that the estimated turning-point approximately 0.5 splits the curve into two regimes, and the two regimes behave so differently. This graph is about the lagged Tobin’s Q’s contribution to the expected investment. Low Q firms (whose potentials are evaluated to be low by the financial market) look rather reluctant to change their future investment plan, or maybe get changed.
Then let us proceed to the surfaces. Check the response from the debta by running
ret$debtaIn an interactive R session, the surface can be rotated and zoomed using the mouse. “vala_y” shows that the y-axis is the Q, and “debta_x” shows that the x-axis is the debt. The tool bar on up-right helps you to rotate, pan, zoom and save the graph.
Note that the transition variable Q is in log scale while debt is not.
It is very clear that low Q firms’ future investment will be affected by the current debt situation. The more debt there is, the less investment there will be. However, this is not the case for high-Q firms, which appear to have better growth prospects and are less sensitive to debt.
The following two living graphs are for the cash flow and the sales.
ret$cfaret$salesIf you use the PSTR package in your research, please cite both the software implementation and the underlying methodology.
Software
Yang, Y. (2026). PSTR: Panel Smooth Transition Regression Modelling. R package version 2.0.0 (first released in 2017). Available at: https://github.com/yukai-yang/PSTR
Methodology
González, A., Teräsvirta, T., van Dijk, D., and Yang, Y. (2005). Panel Smooth Transition Regression Models. EFI Working Paper Series in Economics and Finance, No. 604. Stockholm School of Economics. Revised October 2017. Available at: https://swopec.hhs.se/hastef/papers/hastef0604.pdf
You can obtain the citation information directly from R by running:
citation("PSTR")
#> To cite the PSTR package or its underlying methodology, please use the
#> following references:
#>
#> Yang Y (2026). _PSTR: Panel Smooth Transition Regression Modelling_.
#> R package version 2.0.0 (first released in 2017),
#> <https://github.com/yukai-yang/PSTR>.
#>
#> González A, Teräsvirta T, van Dijk D, Yang Y (2005). "Panel Smooth
#> Transition Regression Models." Technical Report 604, Stockholm School
#> of Economics. Revised October 2017,
#> <https://swopec.hhs.se/hastef/papers/hastef0604.pdf>.
#>
#> To see these entries in BibTeX format, use 'print(<citation>,
#> bibtex=TRUE)', 'toBibtex(.)', or set
#> 'options(citation.bibtex.max=999)'.