Quantile regression models a quantile of the outcome as a function of covariates. Applied researchers use quantile regressions because they allow the effect of a covariate to differ across conditional quantiles. For example, another year of education may have a large effect on a low conditional quantile of income but a much smaller effect on a high conditional quantile of income. Also, another pack-year of cigarettes may have a larger effect on a low conditional quantile of bronchial effectiveness than on a high conditional quantile of bronchial effectiveness.

I use simulated data to illustrate what the conditional quantile functions estimated by quantile regression are and what the estimable covariate effects are. Read more…

\(\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…