Latent variable models for time series: A frequency domain approach with an application to the permanent income hypothesis

The theory of estimation and inference in a very general class of latent variable models for time series is developed by showing that the distribution theory for the finite Fourier transform of the observable variables in latent variable models for time series is isomorphic to that for the observable variables themselves in classical latent variable models. This implies that analytic work on classical latent variable models can be adapted to latent variable models for time series, an implication which is illustrated here in the context of a general canonical form. To provide an empirical example a latent variable model for permanent income is developed, its parameters are shown to be identified, and a variety of restrictions on these parameters implied by the permanent income hypothesis are tested.     

Identification of rational expectations models

This paper deals with the problem of the identification of simultaneous Rational Expectations (RE) models. In the case of RE models with current expectations of the endogenous variables, the necessary and sufficient conditions for the global identification are derived explicitly in terms of the structural parameters and the linear homogenous identifying restrictions. It is shown that in the absence of a priori restrictions on the processes generating the exogenous variables and the disturbances, RE models and general distributed lag models are ‘observationally equivalent’. In the case of RE models with future expectations of the endogenous variables, a general solution that highlights the ‘non-uniqueness’ problem and from which other solutions such as forward or backward solutions can be obtained, is derived. It is shown that untestable and often quite arbitrary restrictions are needed if RE models with future expectations are to be identifiable. Certain order conditions similar to those obtained for the identification of RE models with current expectations are also derived for this case.     

How does the benefit effect vary as unemployment spells lengthen?

This paper investigates how the effect of income while unemployed on the probability of an individual leaving unemployment varies with the length of time that the individual has been unemployed. We examine this question in the context of a variety of alternative econometric models. We extend the Proportional Hazards model with unrestricted baseline hazard to one in which there are unrestricted effects of a subset of the explanatory variables and also consider models that can be estimated as series of binary response models. The proportional hazard restrictions are rejected for the sample of British unemployed men analysed and in the binary sequence framework Logit and Probit models based on symmetric distributions dominate (in likelihood terms) the Extreme Value form model implied by extension of the Proportional Hazards formulation. Logit models with a flexible form for the duration dependence which also incorporate unobserved heterogeneity in a flexible way are estimated. The results for all formulations indicate a rapidly declining effect of unemployment income as a spell lengthens, with no significant effect for the long-term unemployed. The preferred specifications which allow for omitted heterogeneity indicate no significant effect after about 5 months, and this result is robust to the inclusion or exclusion of previous labour-market experience variables and to the choice of mixing distribution.     

A test of non-identifying restrictions and confidence regions for partially identified parameters

We propose an easily implementable test of the validity of a set of theoretical restrictions on the relationship between economic variables, which do not necessarily identify the data generating process. The restrictions can be derived from any model of interactions, allowing censoring and multiple equilibria. When the restrictions are parameterized, the test can be inverted to yield confidence regions for partially identified parameters, thereby complementing other proposals, primarily Chernozhukov et al. [Chernozhukov, V., Hong, H., Tamer, E., 2007. Estimation and confidence regions for parameter sets in econometric models. Econometrica 75, 1243–1285].     

Identification and information in monotone binary models

This paper considers binary response models where errors are uncorrelated with a set of instrumental variables and are independent of a continuous regressor v$v$, conditional on all other variables. It is shown that these exclusion restrictions are not sufficient for identification and that additional identifying assumptions are needed. Such an assumption, introduced by Lewbel [Semiparametric qualitative response model estimation with unknown heteroskedasticity or instrumental variables. Journal of Econometrics 97, 145–177], is that the support of the continuous regressor is large, but we show that it significantly restricts the class of binary phenomena which can be analysed. We propose an alternative additional assumption under which β$\beta$ remains just identified and the estimation unchanged. This alternative assumption does not impose specific restrictions on the data, which broadens the scope of the estimation method in empirical work. The semiparametric efficiency bound of the model is also established and an existing estimator is shown to achieve that bound. The efficient estimator uses a plug-in density estimate. It is shown that plugging in the true density rather than an estimate is inefficient. Extensions to ordered choice models are provided.     

The lag structure of option price

The price of common stock warrants do not adjust immediately to changes in common stock prices. This lag is inconsistent with the ‘efficient market’ hypothesis. Based on daily closing prices this lag was measured and found to be a combination of the adjustment to stock price and to the adjustment of ‘other’ variables, i.e., positive serially correlated disturbance terms. A single equation model simultaneously estimating the parameters of the serial correlation and the coefficients of the lagged stock price indicate a substantial deviation from efficiency. Various simple strategies designed to exploit this lag are then tested.     

Testing Steady‐State Restrictions of Linear Rational Expectations Models when Data are Highly Persistent*

Steady‐state restrictions are commonly imposed on highly persistent variables to achieve stationarity prior to confronting rational expectations models with data. However, the resulting steady‐state deviations are often surprisingly persistent indicating that some aspects of the underlying theory may be empirically problematic. This paper discusses how to formulate steady‐state restrictions in rational expectations models with latent forcing variables and test their validity using cointegration techniques. The approach is illustrated by testing steady‐state restrictions for alternative specifications of the New Keynesian model and shown to be able to discriminate between different assumptions on the sources of the permanent shocks.     

Partial knowledge restrictions on the two-stage threshold model of choice

In the context of the two-stage threshold model of decision making, with the agent’s choices determined by the interaction of three “structural variables,” we study the restrictions on behavior that arise when one or more variables are exogenously known. Our results supply necessary and sufficient conditions for consistency with the model for all possible states of partial knowledge, and for both single- and multi-valued choice functions.     

DISENTANGLING DEMAND AND SUPPLY SHOCKS IN THE CRUDE OIL MARKET: HOW TO CHECK SIGN RESTRICTIONS IN STRUCTURAL VARS

Sign restrictions have become increasingly popular for identifying shocks in structural vector autoregressive (SVAR) models. So far there are no techniques for validating the shocks identified via such restrictions. Although in an ideal setting the sign restrictions specify shocks of interest, sign restrictions may be invalidated by measurement errors, data adjustments or omitted variables. We model changes in the volatility of the shocks via a Markov switching (MS) mechanism and use this device to give the data a chance to object to sign restrictions. The approach is illustrated by considering a small model for the market of crude oil. Earlier findings that oil supply shocks explain only a very small fraction of movements in the price of oil are confirmed and it is found that the importance of aggregate demand shocks for oil price movements has declined since the mid 1980s. Copyright © 2013 John Wiley & Sons, Ltd.     

Identification of simultaneous equation models with measurement error

This paper establishes identification conditions for a simultaneous equation model in which some of the exogenous variables are measured with error. It is assumed that observational information is confined to the covariance matrix of the observed variables and that prior information on the structural coefficients and error variances takes the form of zero restrictions. The primary result is an easily-applied assignment condition for checking whether or not there are an adequate number and variety of prior restrictions to identify the structural parameters.     

The semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions

This paper computes the semiparametric efficiency bound for finite dimensional parameters identified by models of sequential moment restrictions containing unknown functions. Our results extend those of Chamberlain (1992b) and Ai and Chen (2003) for semiparametric conditional moment restrictions with identical information sets to the case of nested information sets, and those of Chamberlain (1992a) and Brown and Newey (1998) for models of sequential moment restrictions without unknown functions to cases with unknown functions of possibly endogenous variables. Our results are applicable to semiparametric panel data models and two stage plug-in problems. As an important example, we compute the efficiency bound for a weighted average derivative of a nonparametric instrumental variables regression (NPIV), and find that simple plug-in NPIV estimators are not efficient. We present an optimally weighted, orthogonalized, sieve minimum distance estimator that achieves the semiparametric efficiency bound.     

Estimating the LQAC model with I(2) variables

This paper derives a method for estimating and testing the Linear Quadratic Adjustment Cost (LQAC) model when the target variable and some of the forcing variables follow I(2) processes. Based on a forward-looking error-correction formulation of the model it is shown how to obtain strongly consistent estimates of the structural parameters from both a linear and a non-linear cointegrating regression where first-differences of the I(2) variables are included as regressors (multicointegration). Further, based on the estimated parameter values, it is shown how to test and evaluate the LQAC model using a VAR approach. A simple easy interpretable metric for measuring the model fit is suggested. In an empirical application using UK money demand data, the non-linear multicointegrating regression delivers an economically plausible estimate of the adjustment cost parameter. However, the restrictions implied by the exact LQAC model under rational expectations are strongly rejected and the metric for model fit indicates a substantial noise component in the model. Copyright © 1999 John Wiley & Sons, Ltd.     

An expository critique of the Takayama—Judge models of interregional and intertemporal market equilibrium

In this paper an attempt is made to expound the spatial-temporal multi-product allocation and price equilibrium models originally proposed by Takayama and Judge (1971). Certain restrictive features of these models are highlighted and an alternative specification which overcomes these restrictions and replaces both the single and multiple storage period models of Takayama and Judge is outlined. The enhanced realism of this new model and its relatively simple structure should commend it to regional economists concerned with policy questions involving the production and distribution of a wide range of commodities.     

On the information criterion for selecting regressors

Summary It has been asserted in the past that any Bayesian treatment of the model selection problem in regression using some form of continuous loss structure would always lead to using the largest possible model (Leamer 1979; Chow 1981). We show in this paper that, provided the distinction between the choice of a model and the estimation of its parameters is maintained, the Kullback-Leibler information measure can be used in a Bayesian context to derive a criterion which may lead to parsimony of parameters in regression analysis. The regression models are taken as restrictions of a general class of distributions which includes the truen-variate distribution of the variabley. Separate criteria for the cases of known and unknown variance ofy are obtained. In the limiting situation when prior opinions about the parameters are weak, these criteria reduce to special cases of the generalizedC _p and AIC criteria (Atkinson 1981). Relationship with other criteria is discussed.     

Instrumental variables regressions with uncertain exclusion restrictions: a Bayesian approach

The identification of structural parameters in the linear instrumental variables (IV) model is typically achieved by imposing the prior identifying assumption that the error term in the structural equation of interest is orthogonal to the instruments. Since this exclusion restriction is fundamentally untestable, there are often legitimate doubts about the extent to which the exclusion restriction holds. In this paper I illustrate the effects of such prior uncertainty about the validity of the exclusion restriction on inferences based on linear IV models. Using a Bayesian approach, I provide a mapping from prior uncertainty about the exclusion restriction into increased uncertainty about parameters of interest. Moderate prior uncertainty about exclusion restrictions can lead to a substantial loss of precision in estimates of structural parameters. This loss of precision is relatively more important in situations where IV estimates appear to be more precise, for example in larger samples or with stronger instruments. I illustrate these points using several prominent recent empirical papers that use linear IV models. An accompanying electronic table allows users to readily explore the robustness of inferences to uncertainty about the exclusion restriction in their particular applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Selection,Heterogeneity, and Entry in Professional Markets

We analyze two different cases of entry regulation in professional markets: first, when licensing is a requirement for becoming a professional (lawyers); second, when entry and price restrictions are applied on a geographical basis (pharmacists). Both cases are investigated within a circular model of localized competition and heterogeneous players. The analysis reveals that licensing introduces a selection mechanism which is effective in preventing entry of inefficient players in markets with large ex ante heterogeneity. Furthermore, because in the second case excessive entry is reduced as the degree of heterogeneity increases, our analysis lends support to a policy that simultaneously relaxes entry and price restrictions.     

Dynamic specification tests for dynamic factor models

We derive computationally simple expressions for score tests of misspecification in parametric dynamic factor models using frequency domain techniques. We interpret those diagnostics as time domain moment tests which assess whether certain autocovariances of the smoothed latent variables match their theoretical values under the null of correct model specification. We also reinterpret reduced‐form residual tests as checking specific restrictions on structural parameters. Our Gaussian tests are robust to nonnormal, independent innovations. Monte Carlo exercises confirm the finite‐sample reliability and power of our proposals. Finally, we illustrate their empirical usefulness in an application that constructs a US coincident indicator.     

Are employment decisions based on rational expectations?

Conventional employment functions with partial adjustment to output fitted to quarterly data tend to have positively autocorrelated residuals, to imply implausibly high returns to scale and almost always fail tests for parameter stability. The hypothesis of this paper is that mis-specified expectations are the main cause of these findings and rational and adaptive expectations models are compared. Further, employment is conditioned not on output but on variables which firms can more reasonably take as exogenous. ‘Disequilibrium’ features of labour markets are introduced by making adjustment costs depend upon current and expected labour market tightness.One of the implications of rational expectations is that the revision between points in time t and $\text{t ? 1}$ in the expected value of any variable should be independent of any information available before t and serially uncorrelated. Given a model of a forward looking firm whose hiring decisions are subject to quadratic adjustment costs, an appropriately transformed employment equation can be derived which has a very similar structure to the Koyck transformed employment equation which corresponds to adaptive expectations. Maximum likelihood estimation of the adaptive expectations form gives parameter estimates for quarterly British data for the manufacturing sector which are so unreasonable that this hypothesis can be rejected. Maximum likelihood estimation of the rational expectations form would involve modelling the stochastic processes of all the driving variables. However, conditional upon one parameter, consistent estimates of the remaining parameters can be obtained by OLS and these accord well with economic theory. This is the direct evidence in favour of the rational expectations hypothesis. However, it can also explain why the adaptive expectations form gives such poor results and why conventional employment functions give the unsatisfactory results referred to above. Further, rational expectations provides an explanation for the common finding, particularly in the context of employment and the demand for durable goods, of implausibly low or wrong signed levels effects in more general quarterly time series models with lagged dependent variables.     

Comparison of misspecified calibrated models: The minimum distance approach

This paper proposes several testing procedures for comparison of misspecified calibrated models. The proposed tests are of the Vuong-type (Vuong, 1989, Rivers and Vuong, 2002). In our framework, the econometrician selects values for model’s parameters in order to match some characteristics of data with those implied by the theoretical model. We assume that all competing models are misspecified, and suggest a test for the null hypothesis that they provide equivalent fit to data characteristics, against the alternative that one of the models is a better approximation. We consider both nested and non-nested cases. We also relax the dependence of models’ ranking on the choice of a weight matrix by suggesting averaged and sup-norm procedures. The methods are illustrated by comparing the cash-in-advance and portfolio adjustment cost models in their ability to match the impulse responses of output and inflation to money growth shocks.