\(\def\bfy{{\bf y}}
\def\bfA{{\bf A}}
\def\bfB{{\bf B}}
\def\bfu{{\bf u}}
\def\bfI{{\bf I}}
\def\bfe{{\bf e}}
\def\bfC{{\bf C}}
\def\bfsig{{\boldsymbol \Sigma}}\)In my last post, I discusssed estimation of the vector autoregression (VAR) model,

\begin{align}
\bfy_t &= \bfA_1 \bfy_{t-1} + \dots + \bfA_k \bfy_{t-k} + \bfe_t \tag{1}
\label{var1} \\
E(\bfe_t \bfe_t’) &= \bfsig \label{var2}\tag{2}
\end{align}

where \(\bfy_t\) is a vector of \(n\) endogenous variables, \(\bfA_i\) are coefficient matrices, \(\bfe_t\) are error terms, and \(\bfsig\) is the covariance matrix of the errors.

In discussing impulse–response analysis last time, I briefly discussed the concept of orthogonalizing the shocks in a VAR—that is, decomposing the reduced-form errors in the VAR into mutually uncorrelated shocks. In this post, I will go into more detail on orthogonalization: what it is, why economists do it, and what sorts of questions we hope to answer with it. Read more…

teffects ipw uses multinomial logit to estimate the weights needed to estimate the potential-outcome means (POMs) from a multivalued treatment. I show how to estimate the POMs when the weights come from an ordered probit model. Moment conditions define the ordered probit estimator and the subsequent weighted average used to estimate the POMs. I use gmm to obtain consistent standard errors by stacking the ordered-probit moment conditions and the weighted mean moment conditions. Read more…

\(\newcommand{\betab}{\boldsymbol{\beta}}\)Time-series data often appear nonstationary and also tend to comove. A set of nonstationary series that are cointegrated implies existence of a long-run equilibrium relation. If such an equlibrium does not exist, then the apparent comovement is spurious and no meaningful interpretation ensues.

Analyzing multiple nonstationary time series that are cointegrated provides useful insights about their long-run behavior. Consider long- and short-term interest rates such as the yield on a 30-year and a 3-month U.S. Treasury bond. According to the expectations hypothesis, long-term interest rates are determined by the average of expected future short-term rates. This implies that the yields on the two bonds cannot deviate from one another over time. Thus, if the two yields are cointegrated, any influence to the short-term rate leads to adjustments in the long-term interest rate. This has important implications in making various policy or investment decisions.

In a cointegration analysis, we begin by regressing a nonstationary variable on a set of other nonstationary variables. Suprisingly, in finite samples, regressing a nonstationary series with another arbitrary nonstationary series usually results in significant coefficients with a high \(R^2\). This gives a false impression that the series may be cointegrated, a phenomenon commonly known as spurious regression.

In this post, I use simulated data to show the asymptotic properties of an ordinary least-squares (OLS) estimator under cointegration and spurious regression. I then perform a test for cointegration using the Engle and Granger (1987) method. These exercises provide a good first step toward understanding cointegrated processes. Read more…

For a nonlinear model with heteroskedasticity, a maximum likelihood estimator gives misleading inference and inconsistent marginal effect estimates unless I model the variance. Using a robust estimate of the variance–covariance matrix will not help me obtain correct inference.

This differs from the intuition we gain from linear regression. The estimates of the marginal effects in linear regression are consistent under heteroskedasticity and using robust standard errors yields correct inference.

If robust standard errors do not solve the problems associated with heteroskedasticity for a nonlinear model estimated using maximum likelihood, what does it mean to use robust standard errors in this context? I answer this question using simulations and illustrate the effect of heteroskedasticity in nonlinear models estimated using maximum likelihood. Read more…

When fitting almost any model, we may be interested in investigating whether parameters differ across groups such as time periods, age groups, gender, or school attended. In other words, we may wish to perform tests of moderation when the moderator variable is categorical. For regression models, this can be as simple as including group indicators in the model and interacting them with other predictors.

We naturally have hypotheses regarding differences in parameters across groups when fitting structural equation models as well. When these models involve latent variables and the corresponding observed measurements, we can test whether those measurements are invariant across groups. Evaluation of measurement invariance typically involves a series of tests for equality of measurement coefficients (factor loadings), equality of intercepts, and equality of error variances across groups.

In this post, I demonstrate how to use the sem command’s group() and ginvariant() options as well as the postestimation command estat ginvariant to easily perform tests of measurement invariance. Read more…

I illustrate that exact matching on discrete covariates and regression adjustment (RA) with fully interacted discrete covariates perform the same nonparametric estimation. Read more…

Introduction

In a univariate autoregression, a stationary time-series variable \(y_t\) can often be modeled as depending on its own lagged values:

\begin{align}
y_t = \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \dots
+ \alpha_k y_{t-k} + \varepsilon_t
\end{align}

When one analyzes multiple time series, the natural extension to the autoregressive model is the vector autoregression, or VAR, in which a vector of variables is modeled as depending on their own lags and on the lags of every other variable in the vector. A two-variable VAR with one lag looks like

\begin{align}
y_t &= \alpha_{0} + \alpha_{1} y_{t-1} + \alpha_{2} x_{t-1}
+ \varepsilon_{1t} \\
x_t &= \beta_0 + \beta_{1} y_{t-1} + \beta_{2} x_{t-1}
+ \varepsilon_{2t}
\end{align}

Applied macroeconomists use models of this form to both describe macroeconomic data and to perform causal inference and provide policy advice.

In this post, I will estimate a three-variable VAR using the U.S. unemployment rate, the inflation rate, and the nominal interest rate. This VAR is similar to those used in macroeconomics for monetary policy analysis. I focus on basic issues in estimation and postestimation. Data and do-files are provided at the end. Additional background and theoretical details can be found in Ashish Rajbhandari’s [earlier post], which explored VAR estimation using simulated data. Read more…

We estimate the average treatment effect (ATE) for an exponential mean model with an endogenous treatment. We have a two-step estimation problem where the first step corresponds to the treatment model and the second to the outcome model. As shown in Using gmm to solve two-step estimation problems, this can be solved with the generalized method of moments using gmm.

This continues the series of posts where we illustrate how to obtain correct standard errors and marginal effects for models with multiple steps. In the previous posts, we used gsem and mlexp to estimate the parameters of models with separable likelihoods. In the current model, because the treatment is endogenous, the likelihood for the model is no longer separable. We demonstrate how we can use gmm to estimate the parameters in these situations. Read more…

\(\newcommand{\Eb}{{\bf E}}
\newcommand{\xb}{{\bf x}}
\newcommand{\betab}{\boldsymbol{\beta}}\)Differences in conditional probabilities and ratios of odds are two common measures of the effect of a covariate in binary-outcome models. I show how these measures differ in terms of conditional-on-covariate effects versus population-parameter effects. Read more…

\(\newcommand{\Eb}{{\bf E}}\)The change in a regression function that results from an everything-else-held-equal change in a covariate defines an effect of a covariate. I am interested in estimating and interpreting effects that are conditional on the covariates and averages of effects that vary over the individuals. I illustrate that these two types of effects answer different questions. Doctors, parents, and consultants frequently ask individuals for their covariate values to make individual-specific recommendations. Policy analysts use a population-averaged effect that accounts for the variation of the effects over the individuals. Read more…