# Nonlinear Models in Macroeconometrics

## Summary and Keywords

Many nonlinear time series models have been around for a long time and have originated outside of time series econometrics. The stochastic models popular univariate, dynamic single-equation, and vector autoregressive are presented and their properties considered. Deterministic nonlinear models are not reviewed. The use of nonlinear vector autoregressive models in macroeconometrics seems to be increasing, and because this may be viewed as a rather recent development, they receive somewhat more attention than their univariate counterparts. Vector threshold autoregressive, smooth transition autoregressive, Markov-switching, and random coefficient autoregressive models are covered along with nonlinear generalizations of vector autoregressive models with cointegrated variables. Two nonlinear panel models, although they cannot be argued to be typically macroeconometric models, have, however, been frequently applied to macroeconomic data as well. The use of all these models in macroeconomics is highlighted with applications in which model selection, an often difficult issue in nonlinear models, has received due attention. Given the large amount of nonlinear time series models, no unique best method of choosing between them seems to be available.

Keywords: Markov-switching model, nonlinear panel model, nonlinear time series, random coefficient model, smooth transition model, threshold autoregressive model, vector autoregressive model

Introduction

The number of nonlinear models in the macroeconometric literature is large. For the purposes of this article, only parametric time series models are considered. This excludes, among other things, nonlinear cross-sectional models and nonparametric time series models. Nonlinear panel models constitute an exception because they bear rather strong resemblance to some of the nonlinear models discussed here. Deterministic models (with random noise) that have occasionally been applied to macroeconomic time series are not reviewed either. Linear models with breaks may also viewed as nonlinear, but the huge literature of structural breaks is omitted from consideration here.

The focus will be on models that do not contain many theory restrictions on their structure. Univariate models naturally belong to this category, and in the vector case, nonlinear vector autoregressive models will receive most of the attention. Further, all models to be highlighted are conditional mean models. Although models of the conditional variance have also been applied to macroeconomic time series (see, e.g., the seminal contribution by Engle, 1982), they are much more popular in financial applications and are left outside this work.

Since applications of nonlinear vector models to macroeconomic time series have become common later than applications of their univariate counterparts, the former models will receive more attention than the latter. Presentations of nonlinear vector models are accompanied by relevant macroeconomic examples, so the reader can see where and how the models can and have so far been applied.

The following topics will be addressed here: univariate models, dynamic single-equation models, multivariate (vector) models, nonlinear panel models, forecasting with nonlinear models, and model selection.

Univariate Autoregressive and Single-Equation Regression Models

## Switching Regression and Smooth Transition Models

Models with more than one regime have a long history in statistics, time series analysis, and econometrics. The first models were regression models with independent observations. Quandt (1958) considered a switching regression model in which the regression equation, including the error term, switches according to a random switch variable. As an example Quandt used a consumption equation in which a switch in the regime occurs when the interest rate exceeds a certain value. Let ${y}_{i}$ be the consumption, ${x}_{i}$ the income, and ${j}_{i}$ the interest rate. The model is

for ${j}_{i}\ge \overline{j},$ and

otherwise, where $(a,b)\ne (c,d)$. The independent errors ${\epsilon}_{ki}\sim {N}(0,{\sigma}_{ki}^{2})$, $k=1,2,$ ${\sigma}_{1i}^{2}\ne {\sigma}_{2i}^{2}.$ The author suggested that the parameters of the model be estimated by maximizing the log-likelihood using a grid over the values of ${j}_{i}$ (In this discussion, however, he used time as the switch variable.) Due to discontinuity of the log-likelihood, this is by and large how parameter estimation is carried out in all switching regression models. Testing the hypothesis that there is only a single regime was also considered. Quandt (1958) acknowledged the fact that the true switch-point, if any, is unknown, but time was not yet ripe for a rigorous analysis of this testing problem.

Bacon and Watts (1971) argued that instead of having an abrupt shift from one regime to the other, one could make the transition smooth. Using previous notation, their smooth transition model has the form

where the transition function $G({x}_{i})$ is a bounded continuous function, monotonic in ${x}_{i}.$ The authors used the two-parameter hyperbolic tangent function that is bounded between zero and one but pointed out that many other functions would be equally possible. They assumed the error variance to be constant for all values of $G({x}_{i})$ and adopted a Bayesian approach to estimating the parameters of the model. The application in Bacon and Watts (1971) was not an economic one but had to do with a chemical.

Interestingly, in the econometrics literature, Goldfeld and Quandt (1972, pp. 263–264) independently presented the smooth transition regression model as a solution to the estimation problem in the switching regression model (1) and (2) with switch variable ${j}_{i}$. The idea was to approximate the switch by a smooth continuous function to make the log-likelihood well behaved and maximize the likelihood using standard nonlinear optimization algorithms.

These models were adapted to time series a few years later. The breakthrough in the univariate case, called the threshold autoregressive (TAR) model, came with the paper by Tong and Lim (1980). The model has the form

where $I(A)$ is the indicator function: $I(A)=1$ when $A$ is true, and zero otherwise, ${c}_{0},{c}_{1},...,{c}_{r}$ are threshold parameters, ${c}_{0}=-\infty ,$ ${c}_{r}=\infty $, and ${\mathbf{\text{w}}}_{t}=(1,{y}_{t-1},...,{y}_{t-p}{)}^{\prime}.$ If $r=1,$ the model is linear. Furthermore, ${\mathit{\text{\varphi}}}_{j}=\mathbf{\text{(}}{\varphi}_{0j},{\varphi}_{1j},...,{\varphi}_{pj}{)}^{\prime}$ such that ${\mathit{\text{\varphi}}}_{i}\ne {\mathit{\text{\varphi}}}_{j}$ for $i\ne j,$ and ${\epsilon}_{jt}={\sigma}_{j}{\epsilon}_{t}$ with $\left\{{\epsilon}_{t}\right\}\sim $ iid $(\mathrm{0,1}),$ and ${\sigma}_{j}>0,$ $j=1,...,r$. The TAR model is called self-exciting when the threshold variable is a lag of ${y}_{t}$ as in (4). In many economic applications the TAR model is assumed to have two regimes:

A comprehensive account of the model and its statistical properties can be found in Tong (1990).

The autoregressive counterpart of the smooth transition model (3) was introduced by Chan and Tong (1986). The smooth transition autoregressive (STAR) model is

where, following previous notation, ${\epsilon}_{jt}=\sigma {\epsilon}_{t}$ with $\left\{{\epsilon}_{t}\right\}\sim $ iid $(\mathrm{0,1}),$ and $\sigma >0.$ The transition function $G({y}_{t-d})$ of Chan and Tong (1986) is the cumulative distribution function of the standard normal variable. The logistic function introduced by Maddala (1977, p. 396) has become the most popular choice in the literature. Teräsvirta (1994) suggested pairing the logistic function

with the exponential function

previously used in a slightly more restricted form in the exponential autoregressive (EAR) model by Haggan and Ozaki (1981). When $\gamma \to \infty $ in (7), the model converges to a two-regime TAR model, whereas the EAR model or the more general ESTAR model (6) with (8) becomes linear. Teräsvirta (1994) discussed the choice between the logistic and the exponential transition function and STAR model specification more generally as well. Jansen and Teräsvirta (1996) suggested another variant of the logistic function that is close to the exponential transition function but contains one parameter more than the latter. A STAR model with this transition function converges to a special case of a TAR model with three regimes when $\gamma \to \infty $ The LSTAR model may be generalized to a multiple-transition model as in van Dijk and Franses (1999), but many economic applications rely on a single-transition model.

A few macroeconomic time series such as interest rates and unemployment rate are rather persistent. Many authors tend to model them as random walks without drift. Lanne and Saikkonen (2002) provided a useful nonlinear alternative that mimics the behavior of these series but can be stationary. Their model is a TAR model in which only the intercept is switching:

where $\mathit{\text{\varphi}}=({\varphi}_{1},...,{\varphi}_{p}{)}^{\prime}$ and Given that the intercept remains bounded, this model has the same stationarity conditions as the linear AR model. The authors fitted their model to two monthly time series: the Swiss Franc euro exchange rate and a UK treasury bill rate.

The argument in the indicator function in (5) may be replaced by an unobservable discrete stochastic variable ${s}_{t}$ that obtains $r$ different values ${s}_{t}\in \{1,...,r\},$ say, and has a (typically first-order) Markov structure. The transition (or staying if $i=j$) probabilities

determine the probability for the process to switch from regime $i$ to regime $j$ at time $t$ and are parameters to be estimated. The resulting model is the Markov switching or hidden Markov autoregressive (MSAR) model. It may be written as follows:

with ${\varphi}_{i}\ne {\varphi}_{j}$ for $i\ne j,$ where ${\epsilon}_{jt}=\sigma {\epsilon}_{t}$ with $\left\{{\epsilon}_{t}\right\}\sim $ $\text{}iid\text{}{N}(\mathrm{0,1})$. Douc, Moulines, and Rydén (2004) considered this model and proved consistency and asymptotic normality for the maximum likelihood estimators of its parameters. Maximum likelihood estimation of the MS regression model was already studied by Lindgren (1978).

Hamilton (1989) introduced a different MSAR model. The latent variable has the same values and transition probabilities as (11) with (10) but a different structure:

where $\mu (i)\ne \mu (j)$ for all $i\ne j.$ From $\left(12\right)$ it is seen that the model bears similarity to (9) in that only the intercept is switching. If $r=2,$ as is often the case in macroeconomic applications, the latent switching intercept can obtain ${2}^{p+1}$ different values.

## Time-Varying Parameter Models and Bayesian Techniques

Another nonlinear model may be constructed from a linear AR model by making its coefficients random. An early example is the autoregressive model

where ${\mathit{\text{\varphi}}}_{1t}=({\varphi}_{1t},...,{\varphi}_{pt}{)}^{\prime}$ and ${\epsilon}_{t}\sim $ iid ${N}(0,{\sigma}^{2}),$ see Andel (1976) and Nicholls and Quinn (1982). The time-varying elements ${\varphi}_{it}\sim $ iid(${\varphi}_{i},{\sigma}_{i}^{2})$. The survey by Swamy and Tavlas (1995) concentrates on random coefficient regression models and contains a number of macroeconomic examples. In many macroeconomic applications, the random coefficients are instead made persistent by assuming that they follow a random walk:

where ${\lambda}_{it}\ge 0,$ $i=1,...,p$. Furthermore, ${\nu}_{it}\sim $ iid ${N}(0,{\sigma}^{2}),$ $i=1,...,p,$ and independent of ${\epsilon}_{t}.$ When ${\lambda}_{it}=0$ for all $i$ and $t$, (13) is a linear autoregressive model. In order to prevent $\left\{{y}_{t}\right\}$ from exploding during the sample period, the parameters ${\lambda}_{it}$ have to be small. Koop and Potter (2001) showed how TVP-AR models are estimated using suitably chosen prior distributions and numerical techniques. Their Bayesian approach also allows model comparisons using Bayes factors.

Bayesian methods also enable simultaneous comparisons of large numbers of time series models. Koop and Potter (2000) discuss building nonlinear models with these methods and apply them to two macroeconomic series, the growth rate of the quarterly real US GDP, 1954(1)–1987(4), and the annual British industrial production index, 1700–1992. They define 11 classes of models: linear models, linear models with one or two structural breaks, linear models with one or two outliers, and two- and three-regime TAR models. In addition, they specify each model with a different number of lags and with homoskedastic or heteroskedastic errors. The idea is to compute the posterior model probability for every model and find out the most probable model or models. In the case of the US GDP, this approach favors a model with a single structural break and heteroskedastic errors, whereas a three-regime TAR model with homoskedastic errors has the highest posterior probability for the industrial production index. Since publication of Koop and Potter (2000), increasing computational power has no doubt increased the attraction of model comparisons of this kind.

## Other Applications

Univariate STAR and TAR models have been applied to a large number of macroeconomic variables, including industrial production, unemployment, and inflation, to name perhaps the most important ones. The main reason for nonlinearity has been asymmetry. For example, dynamic behavior of the growth rate of industrial production has in many countries been different during recessions and expansions. Early examples of this can be found in Teräsvirta and Anderson (1992), who fitted STAR models to quarterly series of growth rates of industrial production in various countries. Similarly, the MSAR model has been applied to characterizing business cycles, in which case the latent variable represents the phase of the cycle—see, for example, Montgomery, Zarnowitz, Tsay, and Tiao (1998) for (11) with (10) and Hamilton (1989) for (12). In macroeconomic applications, the number of regimes in MSAR models is typically chosen a priori and not determined from the data.

Dynamic Single-Equation Regression Models

Augmenting a univariate nonlinear autoregressive model by exogenous variables leads to a dynamic nonlinear regression model. To give an example, the smooth transition regression (STR) model is obtained as

where ${\mathit{\text{\psi}}}_{1}^{\prime}\ne {\mathit{\text{\psi}}}_{2}^{\prime},$ ${\mathbf{\text{x}}}_{t}=({x}_{1t},...,{x}_{kt}{)}^{\prime}$ is a vector of at least weakly exogenous variables, $G({s}_{t})$ is (e.g.,) a logistic transition function, and ${s}_{t}$ is a stationary transition variable. Possibilities include ${s}_{t}={x}_{jt}$ and ${s}_{t}={y}_{t-d},$ but ${s}_{t}$ may also be an exogenous variable not in ${\mathbf{\text{x}}}_{t}.$ STR models have been applied to modeling money demand in Germany and the United Kingdom. For an application to the long annual UK money demand series, originally considered and modeled by Ericsson, Hendry, and Prestwich (1998), see Teräsvirta and Eliasson (2001). A comprehensive model-building strategy for STR models is discussed in Teräsvirta (1998).

Vector Nonlinear Models

Univariate time series models, linear and nonlinear, can be used for forecasting, but describing relationships between macroeconomic variables requires multivariate models, unless exogeneity assumptions can be made. Many of the models fitted to macroeconomic time series are general time series models that have found application in a wide range of areas. They typically nest a standard linear vector autoregressive (VAR) model and are generalizations of corresponding univariate models. A few frequently applied nonlinear VAR models will be discussed in this section.

## Disequilibrium Models

Before considering VAR models, however, the focus will be on a family of models arising from economic theory propositions. There exist situations in economics in which markets do not clear, that is, the ex ante demand and supply quantities cannot be equated, that is, the market is in disequilibrium. For example, a disequilibrium in labor markets may be due to wages that do not adjust downwards. Government-controlled apartment rents are another example of this type of disequilibrium. Further, credit rationing may create a market in disequilibrium: there may be excess demand because the banks may not lend money to firms that they consider too risky to be recipients of loans.

Fair and Jaffee (1972) were the first to define a general disequilibrium model. It contains both a demand and a supply equation. Borrowing the notation in Teräsvirta, Tjøstheim, and Granger (2010, Chapter 2), the demand equation equals

where ${D}_{t}$ is the quantity demanded at time $t,$ ${\mathbf{\text{x}}}_{t}^{D}$ is the vector of variables, except the price, affecting the demand, ${p}_{t}$ is the price at time $t,$ ${\alpha}_{1}<0$ (the price has a negative effect on demand), and ${\epsilon}_{t}^{D}$ is an error term. The supply equation is

where ${S}_{t}$ is the quantity supplied at time $t,$${\mathbf{\text{x}}}_{t}^{S}$ is the vector of variables, other than the price, affecting the supply, ${\beta}_{1}>0$ (the price has a positive effect on supply), and ${\epsilon}_{t}^{S}$ is an error term. When ${D}_{t}\ne {S}_{t},$ only the smaller one of the two quantities is observed. This is indicated by completing (14) and (15) by the “min-condition” for the observed quantity:

The resulting system defined by (14), (15), and (16) is strongly nonlinear. It does not nest a linear system.

Fair and Jaffee (1972) applied the model to the demand and supply of housing starts in the United States. The number of housing starts is a nonstationary variable, and both the demand and supply equations contain a time trend to reflect different nonstationarities in these two equations. Estimation is carried out by quantifying the difference between demand and supply as follows:

This means that the min-condition is replaced by (17). This leads to a single-equation switching regression model

where the switch is instantaneous (observed at the same time as ${p}_{t}$) and the switch-point known. The error process switches as well. An increase in price (in the application a lagged mortgage interest rate) decreases demand for housing starts, whereas a decrease increases the supply.

For maximum likelihood estimation of disequilibrium models with the min-condition, see Maddala and Nelson (1974). For a Bayesian approach and more discussion about the model, see Bauwens, Lubrano, and Richard (1999, Section 8.6).

## Vector Smooth Transition Regression Model

The first VAR model to be considered here is the logistic vector STR (LVSTR) model. It is a generalization of the single-equation STR model to the vector case. Following Hubrich and Teräsvirta (2013), the LVSTR model of order $p$ may be defined as follows:

where ${\mathbf{\text{y}}}_{t}$ is an $m\times 1$ vector of stationary variables, ${\mathbf{\text{x}}}_{t}$ is an $n\times 1$ vector of stationary exogenous variables, ${\mathit{\text{\mu}}}_{0}$ and ${\mathit{\text{\mu}}}_{1}$ are $m\times 1$ intercept vectors, ${\mathbf{\text{\Phi}}}_{j}$ and ${\mathbf{\text{\Psi}}}_{j},$$j=1,...,p,$ are $m\times m$ parameter matrices, and $\mathbf{\text{\Gamma}}$ and $\mathbf{\text{\Xi}}$ are $m\times n$ parameter matrices. Each row of the composite matrix $[{\mathit{\text{\mu}}}_{1},{\mathbf{\text{\Psi}}}_{1},...,{\mathbf{\text{\Psi}}}_{p},\mathbf{\text{\Xi}}]$ has to contain at least one nonzero element. The $m\times m$ transition matrix $\mathbf{\text{G}}(\mathit{\text{\gamma}},\mathbf{\text{c}};{\mathbf{\text{s}}}_{t})$ has the following form:

where ${s}_{it},$$i=1,...,m,$ are stationary transition variables. The error vector ${\mathit{\text{\epsilon}}}_{t}\sim \text{iid}(\mathbf{\text{0}},\mathbf{\text{\Sigma}})$ where $\mathbf{\text{\Sigma}}>0$. When ${G}_{j}({\gamma}_{j},{\mathbf{\text{c}}}_{j},{s}_{jt}),$$j=1,...,m,$ are standard logistic functions,

and $\mathbf{\text{\Gamma}}=\mathbf{\text{\Xi}}=\mathbf{\text{0,}}$ the model (18) is stable if both $|{\mathbf{\text{I}}}_{m}-{\sum}_{j=1}^{p}{\mathbf{\text{\Phi}}}_{j}{z}^{j}|\ne 0$ and $|{\mathbf{\text{I}}}_{m}-{\sum}_{j=1}^{p}({\mathbf{\text{\Phi}}}_{j}+{\mathbf{\text{\Psi}}}_{j}){z}^{j}|\ne 0$ for $\left|z\right|\le 1.$

A special case found in many applications is the one in which (19) is simplified to

where a single transition function controls the shift in all equations. Camacho (2004) considered this model and devised a modeling strategy for it. Replacing the transition function in (21) by $I({s}_{t}\le c)$ and setting $\mathbf{\text{\Gamma}}=\mathbf{\text{\Xi}}=\mathbf{0}$ in (18) yields the two-regime vector threshold autoregressive (VTAR) model by Tsay (1998).

Neither LVSTR nor the VTAR model is identified when the data-generating process is linear. To avoid the estimation of unidentified models, it follows that linearity has to be tested before fitting either of the two models to the data.

Interaction between the real and financial sectors of the economy has become under scrutiny, especially after the financial crisis of years 2007–2008. Schleer and Semmler (2015) apply the LVSTAR model ($\mathbf{\text{\Gamma}}=\mathbf{\text{\Xi}}=\mathbf{\text{0}}$ in (18)) to study this interaction in 11 euro-area countries. They assume that there may be two extreme regimes: a low and a high (financial) stress regime. To study the effects of financial stress to the economy they construct for each country a two-dimensional LVSTAR model with the growth rate and the corresponding ZEW Financial Condition Index (FCI) for the euro area financial conditions as variables. The transition variable ${s}_{t}$ in (21) is a lag of FCI of that country. Linearity is tested and rejected before specifying and estimating a nonlinear model. Generalized impulse response functions (GIRF; see, e.g., Koop, Pesaran, & Potter, 1996 or Teräsvirta et al., 2010, Chapter 15) computed from the estimated model are used to interpret the results. They show that the response to a financial shock is stronger and longer-lasting during financial stress than when the stress is low.

Caggiano, Castelnuovo, and Figueres (2017) consider the effect of policy uncertainty on central macroeconomic variables of the US economy during different phases of the business cycle. The policy uncertainty is measured by an index constructed by Baker, Bloom, and Davis (2016). The variables are the six-term moving average of the monthly growth rate of industrial production, the unemployment rate, the year-on-year CPI inflation, and the federal funds rate. In addition, the model contains a binary policy uncertainty dummy variable based on the uncertainty index. After testing and rejecting linearity, the authors construct an LVSTAR model for these variables. A smoothed and lagged growth rate of the industrial production functions as the transition variable in (21). Even here, GIRF are used to illustrate the results. They show that exogenous policy uncertainty shocks have stronger effects on the economy in recessions than in expansions. See also Caggiano, Castelnuovo, and Groshenny (2014) for an application of the LVSTAR model to estimating the effects of policy uncertainty on the US unemployment rate.

## Vector Smooth Transition Error Correction Model

In the previous section it was assumed that the variables in the model are stationary. Many macroeconomic variables are, however, nonstationary, and some of them “move together,” in which case they may be assumed (or shown) to be linearly cointegrated. Based on this assumption, Gefang (2012) used a vector smooth transition error correction (VSTEC) model to study the money-output relationship. An interesting thing is that she estimates the model using Bayesian techniques. Bayesian methods in general are frequently used in the estimation of nonlinear VAR models because they tend to alleviate numerical problems present in the estimation of some of them.

The logistic VSTEC model in Gefang (2012) is obtained by reparameterizing the LVSTAR model, (18) with $\mathbf{\text{\Gamma}}=\mathbf{\text{\Xi}}=\mathbf{\text{0}},$ and the transition function (21) as follows:

where $\Delta {\mathbf{\text{y}}}_{t}$ is assumed stationary in the mean, ${\mathbf{\text{D}}}_{t}$ contains the deterministic components (in (18) the intercept, in (22) the intercept and the linear time trend), and the $m\times m$ matrix ${\mathbf{\text{\Pi}}}_{i}={\mathbf{\text{A}}}_{i}{\mathbf{\text{B}}}_{i}^{\prime},$ with rank $\left({\mathbf{\text{A}}}_{i}\right)=$ rank $({\mathbf{\text{B}}}_{i})={q}_{i}<m\text{}$ and $i=\mathrm{0,1.}$ Note that the ranks need not be equal. The cointegrating relationships defined in ${\mathbf{\text{B}}}_{i}$ are thus assumed to change with the regime, which would probably complicate the classical specification and estimation procedure quite substantially. Details of how this and other difficulties are handled in the Bayesian framework are discussed in the article. It may be mentioned, however, that to avoid the aforementioned identification problem: (22) is not identified if the true data-generating process is linear, the prior distribution for the slope parameter $\gamma $ is bounded away from zero. There are 18 candidates for the transition variable ${s}_{t},$ and the Bayesian approach allows one to investigate all of them.

The model in the application is a four-variable LSTVAR consisting of the seasonally adjusted industrial production index, the seasonally adjusted M2 money stock, the producer price index for all commodities, and the secondary market rate on three-month treasury bills. The observations are monthly US data from 1959(1) to 2006(12). Bayesian posterior probabilities are calculated for 3,138 models in total. The models with the highest posterior probability have one factor in common: they all suggest that money nonlinearly Granger causes output.

## Vector Threshold Autoregressive Model

As already mentioned, replacing the transition function in (21) by $I({s}_{t}\le c)$ and setting $\mathbf{\text{\Gamma}}=\mathbf{\text{\Xi}}=\mathbf{\text{0}}$ in (18) yields the two-regime vector threshold autoregressive (VTAR) model by Tsay (1998). The author developed a useful strategy for building such models. The application in that paper is to financial series and is therefore not considered here. Galvão (2006) contains an interesting macroeconomic application in which the VTAR model has an extra twist: it also allows for a break in the series. The vector structural break threshold autoregressive (VSBTAR) model is defined as follows:

where ${s}_{t}$ is the threshold variable, $d>0,$ ${\mathit{\text{\epsilon}}}_{t}\sim $ iid $(\mathbf{\text{0}},\mathbf{\text{\Sigma}}),$ ${r}_{1}$ and ${r}_{3}$ are switch-points, and ${t}_{0}$ is a break-point. When ${I}_{t}\equiv 1,$ (23) collapses into a VTAR model, whereas when ${I}_{z}\equiv 1,$ the model is a linear VAR with a break at $t={t}_{0}.$ The standard two-regime VTAR model thus becomes

The structure of (23) resembles that of the single-equation time-varying STAR model of Lundbergh, Teräsvirta, and van Dijk (2003) with the extension that ${r}_{1}\ne {r}_{3}.$ The model is aimed at describing the relationship between economic growth and interest rate spread. The purpose is to investigate the claim that the spread forecasts economic growth in recessions but not in expansions. The possibility that the relationship is changing over time is considered as well.

Specification of VSBTAR models requires care due to the double transition structure. Since the model is only identified under the alternative, Galvão (2006) applies supremum linearity tests. For these tests, see, for instance, Hansen (1996) or Teräsvirta et al. (2010, Section 5.5). The parameters are estimated by conditional least squares or by maximum likelihood conditionally on the switch-points ${r}_{1},{r}_{3}$ and ${t}_{0}.$ A three-dimensional grid is formed for these three parameters, and the (global) optimum of the objective function (log-likelihood or the sum of squared errors) yields the estimates for them.

The VSBTAR model is applied to predicting recessions in the US economy. There exists literature suggesting that interest rate spreads are useful in predicting output growth only when the growth rate is negative but not otherwise. This relationship could be described by a VTAR model. But then, Galvão (2006) also cites research suggesting that spread may have lost its predictive power, which could be investigated by a linear VAR model with breaks. Considering these proposals jointly leads to the bivariate VSBTAR model, whose variables are the output growth and the spread between the long- and short-term interest rates. The observations are quarterly from 1953(2) to 2002(4). The model selection procedure supports the choice of the VSBTAR model.

It is not possible here to describe the forecasting procedure or how success in predicting recessions is measured. Galvão (2006) reports that the VSBTAR model performs better than its competitors, VAR and VTAR, in-sample, whereas the latter two models are “more robust,” meaning that they outperform the more complicated VSBTAR model out-of-sample. The results show that the estimate of the break-point in the VSBTAR model is changing when new data become available. This may not be surprising because the model allows exactly one break, and it may be reasonable to expect the most conspicuous shift in parameters to be the one determining the location of the sole break-point.

## Vector Threshold Cointegration

The LVSTEC model (22) was preceded by a vector threshold cointegration (VTC) model that Balke and Fomby (1997) introduced. The VTC model with three regimes can be written as follows:

for $p\ge 2,$ where $\mathbf{\text{\beta}}$ and ${\mathbf{\text{\alpha}}}_{j},$ $j=1,2,3,$ are $m\times 1$ vectors and $\left\{{\mathit{\text{\epsilon}}}_{t}^{(j)}\right\}\text{}$ is a sequence of independent but not identically distributed vectors with mean $\mathbf{\text{0}}$ and covariance matrix ${\mathbf{\text{\Sigma}}}_{j}.$ In this model, the cointegrating relationship ${s}_{t}={\mathit{\text{\beta}}}^{\prime}{\mathbf{\text{y}}}_{t}$ and the switch variable equals ${s}_{t-1}.$ If ${c}_{1}<0$ and ${c}_{2}>0$ (assuming as before that ${c}_{0}=-\infty $ and ${c}_{3}=+\infty )$ and, furthermore, ${\mathit{\text{\alpha}}}_{2}=\mathbf{\text{0,}}$ the model describes a situation in which there is a band around the equilibrium ${s}_{t}=0$ such that within the band no adjustment toward the equilibrium takes place. When $p=1,$ the lags of $\Delta {\mathbf{\text{y}}}_{t}$ vanish from the model. Saikkonen (2008) considered (25) assuming ${\mathbf{\text{\Psi}}}_{k}^{(j)}={\mathbf{\text{\Psi}}}_{k}$ and ${\mathit{\text{\epsilon}}}_{t}^{(j)}={\mathit{\text{\epsilon}}}_{t},$ $j=\mathrm{1,2,3};$ $k=1,...,p-1.$ Furthermore, he also discussed the case in which instead of three distinct regimes the transition from one extreme to the other is smooth and described by two logistic transition functions.

The VTC model is often applied to describing the relationship between two interest rates. The argument is that transaction costs prevent the adjustment inside a band. In these cases often $\mathit{\text{\beta}}\mathbf{\text{=}}\mathbf{\text{(}}1,-1\mathbf{\text{)}}\mathbf{\text{.}}$ Besides, it may be assumed that the band is symmetric around zero: ${c}_{1}=-{c}_{2}$ in (25). Bec and Rahbek (2004) studied a pair of short-term and long-term German interest rates using a VTC model. Their univariate tests rejected the unit root hypothesis against a stationary threshold alternative for both series. The authors then fitted a univariate TAR model to ${\mathbf{\text{\beta}}}^{\prime}{\mathbf{\text{y}}}_{t}$ and, using a supremum linearity test, found that linearity was rejected. From this they concluded that the two series may be nonlinearly cointegrated and fitted a VTC model to them.

Anderson (1997) suggested another nonlinear adjustment mechanism to consider the treasury bill market. The model is (25) except for two differences. First, the indicator function in (25) is replaced by the exponential transition function (8) with $c=0\text{}($ as defined in the paper, ${s}_{t}=0$ is the equilibrium point at time $t$), second, the delay $d=1,$ and third, ${\mathit{\text{\epsilon}}}_{t}\sim $ iid ($\mathbf{\text{0}}$, $\mathbf{\text{\Sigma}}).$ This makes the adjustment smooth and nonlinear such that the drift toward the equilibrium first increases and becomes constant when $\left|{s}_{t-1}\right|$ becomes sufficiently large. The argument for this transition function is that different treasury bill owners face different transaction costs, in which case a sharp band is not a suitable description of the aggregate.

It may be mentioned that the (bivariate) vector threshold error correction model is substantially generalized by Cai, Gao, and Tjøstheim (2017), who apply their model to characterizing the relationship between the US federal funds rate controlled by the Federal Reserve and the three-month treasury bill rate. The results suggest that “the Federal Reserve tends to adjust the federal funds rate as a response to the market interest rates.” A detailed treatment of the model is not possible here.

## Vector Markov Switching Autoregressive Models

Like the univariate smooth transition or threshold autoregressive models, the univariate Markov switching model can also be generalised to a vector model. The two-regime vector Markov switching autoregressive (VMSAR) model may be obtained by replacing the indicator function in (24) by $I({s}_{t}=i)$, where ${s}_{t}$ is latent and $i=\mathrm{1,2.}$ The dynamic behaviour of the latent variable is defined as in the univariate case. For a review, see Krolzig (1997).

VMSAR models are quite popular in macroeconomics. As their univariate counterparts, they are suitable for situations in which it can be assumed (sometimes because of lack of information) that the probability of switching regimes is constant over time and does not depend on any observable indicator variable. Like the VTAR model, the VMSAR model nests a linear VAR model. It has the same property as the VTAR model: the VMSAR model is not identified when the true model is a linear VAR.

Warne and Vredin (2006) considered whether the unemployment is more (or less) volatile when inflation is high than when it is low. This is done for three countries: the United States, the United Kingdom, and Sweden. We choose the US model to illustrate their work. The authors begin by constructing a theory model and continue by deriving its time series counterpart, a bivariate VAR model. Since there is a possibility that the two series are cointegrated, they first estimate a linear error correction VAR model. Misspecification tests for the model based on monthly US data from 1959(1) to 1998(12) show that the estimated model is not satisfactory: the errors are autocorrelated and seem to contain conditional heteroskedasticity.

The authors next consider a bivariate VMS error correction (VMS-EC) model for ${\mathbf{\text{y}}}_{t}=({y}_{INF,t},{y}_{U,t}{)}^{\prime}.$ It has the following form:

where ${\mathit{\text{\epsilon}}}_{t}\sim $ iid $(\mathbf{\text{0}},\mathbf{\text{\Sigma}}).$ The cointegrating vector $\mathit{\text{\beta}}\mathbf{\text{=}}\mathbf{\text{(}}1,-{\beta}_{U}),$ where $-{\beta}_{U}$ is the coefficient of ${y}_{U,t-1}.$ Before fitting (26) to the data, linearity (one regime) is tested against two regimes using the test by Carrasco, Hu, and Ploberger (2014) and rejected. It should be noted that in most macroeconomic applications linearity is not tested but the number of regimes is simply assumed to be known. Testing is important, however, for the reason already mentioned: the MS-AR model, like the TAR and STAR models, is not identified when the true relationship is linear.

It is common to estimate the cointegrating relationship from the linear VAR and keep it fixed in (26). Warne and Vredin (2006) instead construct a grid for ${\beta}_{U}$ and estimate the other parameters conditionally on values of ${\beta}_{U}$ in the grid using the EM algorithm. The estimated equations are evaluated using misspecification tests in Hamilton (1996) and their vector generalizations and found adequate. The authors estimate a $95\%$ confidence interval for ${\beta}_{U}$ (its estimate equals ${\widehat{\beta}}_{U}=0.038)$ by using a grid as explained in the paper. The interval contains zero, and the conclusion is that inflation is actually stationary, whereas there may be a stochastic trend in unemployment. The estimated regimes are interpreted as low- and high-inflation ones, and the outcomes pertaining to the original research question are discussed.

When the dimension of the model increases, estimation of VMSAR models often becomes numerically very demanding. In such a situation, Bayesian methods may help. It is not possible to discuss Bayesian VMSAR models in detail here, but a reference is made to the paper by Sims, Waggoner, and Zha (2008). The authors show how parameter restrictions in the transition matrix $\mathbf{\text{P}}=[{p}_{ij}],$ where ${p}_{ij}\text{}$ is defined as in (10) make the VMSAR model a flexible and applicable tool in many situations. For example, it may be used to model structural shifts as well as incremental changes in parameters over time.

Sims et al. (2008) discuss the issue of constructing prior distributions for parameters. They consider the case in which both the mean and the variance of the process are changing over time. They develop a new estimation method that is computationally more efficient than the widely used Monte Carlo EM method. The application in the paper is to the trivariate vector series consisting of the logarithm of the GDP, an inflation variable, and the federal funds rate. Nine different models are specified and estimated. The ones with three or four regimes in which only the variance is switching are found to have the best fit, measured by the marginal data density, a concept defined in the paper.

## Vector Random Coefficient Autoregressive Models

Assuming coefficients of a linear VAR model to be random generates another family of nonlinear models. Consider the following VAR model:

where ${\mathbf{\text{y}}}_{t}$ is an $m\times 1$ vector, ${\mathbf{\text{\Phi}}}_{jt},$ $j=1,...,p,$ are stochastic $m\times m$ parameter matrices, and ${\epsilon}_{t}\sim $ iid ($\mathbf{\text{0}},\mathbf{\text{\Sigma}}$). Define the $m\times pm$ matrix ${\mathbf{\text{\Phi}}}_{t}=({\mathbf{\text{\Phi}}}_{1t},...,{\mathbf{\text{\Phi}}}_{pt})$ and vectorize it into a $p{m}^{2}$-vector ${\mathit{\text{\varphi}}}_{t}=$ vec(${\mathbf{\text{\Phi}}}_{t})$. Nicholls and Quinn (1981a, 1981b) assumed that ${\mathbf{\text{\varphi}}}_{\mathit{\text{t}}}\sim $ iid($\mathit{\text{\varphi}}\mathbf{\text{,}}\mathbf{\text{\Omega}}\mathbf{\text{)}}$ and considered stationarity conditions and asymptotic properties of least squares estimators of parameters of this vector random coefficient autoregressive (VRCAR) model. More recently, as in the univariate case, in economic applications the focus has been on (27) such that ${\mathit{\varphi}}_{t}={\mathit{\text{\varphi}}}_{t-1}+{\mathit{\text{\nu}}}_{t}$ with ${\mathit{\text{\nu}}}_{t}\sim $ iid ${N}(\mathbf{\text{0,}}\mathbf{\text{\Omega}}).$ Furthermore, cov(${\mathit{\epsilon}}_{t},{\mathit{\text{\nu}}}_{t})=\mathbf{\text{\Lambda}}.$ This means that the sequence {${\mathit{\text{\varphi}}}_{t}\},$ instead of being iid, is a random walk without drift. The paths of the individual coefficients may diverge and make $\left\{{\mathbf{\text{y}}}_{t}\right\}$ an explosive sequence.

Cogley and Sargent (2001) apply this VRCAR model to studying the relationship between inflation, unemployment, and the real interest rate. The approach is Bayesian, and the values of ${\mathit{\text{\varphi}}}_{t}$ are obtained by simulation after postulating prior distributions for the starting-value ${\mathit{\text{\varphi}}}_{0}$ and the hyperparameters $\mathbf{\text{\Sigma}}\mathbf{\text{,}}$ $\mathbf{\text{\Omega}}$ and $\mathbf{\text{\Lambda}}.$ However, since the variance of inflation in this model approaches infinity over time, which, as the authors write, “cannot be optimal for a central bank that minimizes a loss function involving the variance of inflation,” in simulations the draws from the conditional distribution of ${\mathit{\text{\varphi}}}_{t}$ given ${\mathit{\text{\varphi}}}_{t-1}$ and $\mathbf{\text{\Omega}}$ leading to explosive roots of the lag polynomial of (27) at time $t$ are discarded. In fact, the variance of the other two variables approaches infinity as well. This restriction implies that $\left\{{\mathbf{\text{y}}}_{t}\right\}$ is persistent but stationary with unknown dynamic properties. Primiceri (2005), applying a similar (but not identical) model, does not impose such a restriction, the argument being that the observation period is so short that the coefficients do not have time to explode. Whether or not this happens also depends on the properties of $\mathbf{\text{\Omega}}\mathbf{\text{.}}$

The argument for fitting a VRCAR model to this dataset is that the dynamic relationship between the variables in ${\mathbf{\text{y}}}_{t}$ is constantly changing, and a model with random walk parameters is therefore better suited for characterizing the relationship than, say, a VAR model with constant parameters. In the present example, the reason for fitting this reduced form (in Primiceri’s case structural) model to the data is that its time-varying parameter estimates are expected to provide information about changes in monetary policy over the years.

In applications of nonlinear models that nest a linear VAR, testing linearity against these models is possible and, as has been discussed, even necessary. There does not seem to exist a test of a linear VAR against the type of parameter change in the VRCAR model of Cogley and Sargent (2001). (There does not seem to exist a test in which the VRCAR model would be the null hypothesis.) The test by Nyblom (1989) comes closest as its alternative is that the parameter vector is a random walk. In Cogley and Sargent (2005) the test is carried out and the null hypothesis is not rejected. Some tests of the linear VAR against a structural break yield the same result. The authors argue that these tests have low power against their VRCAR. Their conclusion is that “a failure to reject should not be construed as an embarrassment to time-varying parameter models.” A practical conclusion would be that since the null hypothesis is not rejected, the linear VAR ought to be preferred to the computationally more complicated nonlinear VRCAR model.

The VRCAR models of Cogley and Sargent (2005) and Primiceri (2005) also contain a time-varying error covariance matrix, based on stochastic volatility. Since the focus here is on the conditional mean models and because of space restrictions, this extension is not considered. The aforementioned tests, however, are performed under the assumption that the error covariance matrix is constant over time.

Nonlinear Panel Models

Although this article has concentrated on pure time series models, there is an important related area worth mentioning, namely the nonlinear panel models. In these models the time dimension that contains nonlinearity is completed with cross sections. There exist two popular nonlinear panel models: the panel threshold regression (PTR) model by Hansen (1999) and the panel smooth transition regression (PSTR) model by González, Teräsvirta, van Dijk, and Yang (2017). In the multi-threshold form the PTR model has the following representation:

for $i=1,...,N,$ where ${y}_{it}$ is a scalar, ${\mathbf{\text{x}}}_{it}$ is a vector of regressors for the cross-sectional unit $i,$ the nonzero vectors ${\varphi}_{j}\ne {\mathit{\text{\varphi}}}_{k}$ for all $j\ne k,$ ${s}_{it}$ is the threshold variable for this unit, and the error term ${\epsilon}_{it}\sim $ iid($0,{\sigma}^{2}).$ The parameters ${c}_{0}=-\infty $ and ${c}_{r}=\infty $ as in (4). Hansen (1999) considers linearity testing, determining $r,$ and estimation by nonlinear least squares.

The PSTR model is similar to (28) and is defined by the following equation:

where ${\lambda}_{t}$ is a time-specific variable, for example, a trend, common to all units $i,$ and ${\epsilon}_{it}\sim $ iid($0,{\sigma}_{i}^{2}),$ $i=1,...,N,$ that is, heteroskedasticity is allowed in cross sections. The transition function equals

where different transitions can have different transition variables for each $i$ and, typically, $K=1$ or $K=2.$ Modeling problems include determining $r$ and $K,$ and they are discussed in the article.

Both models have been applied to macroeconomic problems and datasets. For example, the relationship between economic growth and inflation (“the growth-inflation nexus”) can be perceived as a nonlinear phenomenon. Applications of nonlinear panel models to studying it include Espinoza, Leon, and Prasad (2011), Omay and Öznur Kan (2010), and Seleteng, Bittencourt, and van Eyden (2013). The Feldstein-Horioka puzzle of positive saving-investment correlations may be regarded as another one. Fouquau, Hurlin, and Rabaud (2008) considered the relationship between these two variables for 24 OECD countries annually from 1960 to 2000 using the PSTR model. Additional references to published work can be found in González et al. (2017).

Forecasting With Nonlinear Models

Many nonlinear models, univariate ones in particular, are being constructed and estimated for forecasting. Forecasting nonlinear (macro)economic time series is therefore an important topic. For space reasons, however, it cannot be taken up here. Suffice it to say that forecasting nonlinear series is different from linear ones in that for many nonlinear models, even point forecasts cannot be obtained analytically. For a discussion of forecasting techniques using nonlinear models, see Teräsvirta et al. (2010, Chapter 14). For a general survey on forecasting with nonlinear models, see Teräsvirta (2006a). Useful examples of forecasting macroeconomic variables with nonlinear models such as the STAR and neural network models, see Stock and Watson (1999) and Teräsvirta, van Dijk, and Medeiros (2005). Neural network models have not been considered here. White (2006) provides an illuminating review focusing on specification and estimation issues of these models.

How to Choose Between Models?

As a macroeconometric model builder has a large amount of nonlinear models to choose from, a natural question to ask is: Which one to choose? This choice already affects another important decision: the choice between the linear and nonlinear model. The reason is that many, albeit not all, linearity tests are tests against a particular nonlinear alternative that nests a linear model. Nonparametric tests constitute an exception. For discussion about this problem, see Teräsvirta et al. (2010, Section 5.8).

An ideal situation is the one in which economic theory determines the choice. Disequilibrium models considered earlier may serve as an example. This is not very common, however. For example, most nonlinear models considered here are extensions of linear VAR models and as such not restricted by economic theory. Using model selection criteria such as AIC or BIC has the advantage that they allow comparisons between nonnested models. Bayes factors have already been mentioned. Since most models discussed earlier nest a linear model, they can be meaningfully compared only if linearity is first tested and rejected. Comparing unidentified nonlinear models when the data-generating process is linear does not make any sense.

But then, one may argue that in-sample fit is not a reliable criterion because it does not necessarily guarantee adequate out-of-sample performance. Out-of-sample performance of a model may therefore be viewed as a helpful selection criterion. A problem is that a particular nonlinear model may be built for describing a phenomenon that does not occur very frequently. If it fails to occur during the forecasting period, the model may not be considered useful. A sufficiently long forecasting period is therefore needed to discriminate between models. Teräsvirta and Anderson (1992) made this point while comparing forecasting performance of STAR models of industrial production with their linear AR counterparts.

Since nonlinear models are estimated using numerical methods, it may happen that the algorithm used for the purpose does not converge or yields parameter estimates that are deemed unrealistic. In those cases estimation works as a “negative” criterion: it helps model selection in the sense that some clearly inappropriate alternatives can be weeded out.

Acknowledgment

This research has been partly supported by the Center for Research in Econometric Analysis of Time Series (CREATES), funded by the Danish National Research Foundation, Grant No. DNRF 78.

## Further Reading

Due to the large number of nonlinear models applied to macroeconomic problems, many such models have not received the attention they might deserve. For more information, the reader has to turn to other sources. For an overview specifically focussing on macroeconometric time series, see Granger (2001). A survey of disequilibrium models can be found in Maddala (1983, Chapter 10). For an authoritative treatment of univariate TAR models the reader is referred to Tong (1990); see also Tong (2011) and Tsay (1989). Hansen (2011) contains a comprehensive account of economic applications of these models. Teräsvirta et al. (2010) take up a number of nonlinear models not discussed in this work, including nonparametric nonlinear models. Among other things, they also discuss building STAR and TAR models. Nonparametric models play a central role in Fan and Yao (2003). Tjøstheim (1994) also emphasizes nonparametric nonlinear models and methods. Univariate nonlinear models are surveyed in Teräsvirta (2006b) and STAR models in particular in van Dijk, Teräsvirta, and Franses (2002). The volume of Krolzig (1997) on Markov-switching models has already been mentioned. There exists a rather recent survey of vector TAR and STAR models by Hubrich and Teräsvirta (2013). De Gooijer (2017) is a general up-to-date reference to both uni- and multivariate nonlinear time series covering models used in macroeconometrics.

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