A systematic framework for analyzing the dynamic effects of permanent and transitory shocks

This paper proposes a systematic framework for analyzing the dynamic effects of permanent and transitory shocks on a system of n economic variables. We consider a two-step orthogonolization on the residuals of a VECM with r cointegrating vectors. The first step separates the permanent from the transitory shocks, and the second step isolates n?r mutually uncorrelated permanent shocks and r transitory shocks. The decomposition is computationally straightforward and entails only a minor modification to the Choleski decomposition commonly used in the literature. We then show how impulse response functions can be constructed to trace out the propagating mechanism of shocks distinguished by their degree of persistence. In an empirical example, the dynamic responses to the identified permanent shocks have properties similar to shocks to productivity, the real interest rate, and money growth, even though no economic theory was used to achieve the identification. We highlight two numerical issues that could affect any identification of permanent and transitory shocks.Publicad

Understanding and Comparing Factor-Based Forecasts

Forecasting using "diffusion indices" has received a good deal of attention in recent years. The idea is to use the common factors estimated from a large panel of data to help forecast the series of interest. This paper assesses the extent to which the forecasts are influenced by (i) how the factors are estimated and/or (ii) how the forecasts are formulated. We find that for simple data-generating processes and when the dynamic structure of the data is known, no one method stands out to be systematically good or bad. All five methods considered have rather similar properties, though some methods are better in long-horizon forecasts, especially when the number of time series observations is small. However, when the dynamic structure is unknown and for more complex dynamics and error structures such as the ones encountered in practice, one method stands out to have smaller forecast errors. This method forecasts the series of interest directly, rather than the common and idiosyncratic components separately, and it leaves the dynamics of the factors unspecified. By imposing fewer constraints, and having to estimate a smaller number of auxiliary parameters, the method appears to be less vulnerable to misspecification, leading to improved forecasts.

Confidence Intervals for Diffusion Index Forecasts with a Large Number of Predictor

We consider the situation when there is a large number of series, $N$, each with $T$ observations, and each series has some predictive ability for the variable of interest, $y$. A methodology of growing interest is to first estimate common factors from the panel of data by the method of principal components, and then augment an otherwise standard regression or forecasting equation with the estimated factors. In this paper, we show that the least squares estimates obtained from these factor augmented regressions are $\sqrt{T}$ consistent if $\sqrt{T}/N\rightarrow 0$. The factor forecasts for the conditional mean are $\min[\sqrt{T},\sqrt{N}]$ consistent, but the effect of ``estimated regressors' is asymptotically negligible when $T/N$ goes to zero. We present analytical formulas for predication intervals that take into account the sampling variability of the factor estimates. These formulas are valid regardless of the magnitude of $N/T$, and can also be used when the factors are non-stationary. The generality of these results is made possible by a covariance matrix estimator that is robust to weak cross-section correlation and heteroskedasticity in the idiosyncratic errors. We provide a consistency proof for this CS-HAC estimator.Panel data, common factors, generated regressors, cross- section dependence, robust covariance matrix

Understanding and Comparing Factor-Based Forecasts

Forecasting using `diffusion indices' has received a good deal of attention in recent years. The idea is to use the common factors estimated from a large panel of data to help forecast the series of interest. This paper assesses the extent to which the forecasts are influenced by (i) how the factors are estimated, and/or (ii) how the forecasts are formulated. We find that for simple data generating processes and when the dynamic structure of the data is known, no one method stands out to be systematically good or bad. All five methods considered have rather similar properties, though some methods are better in long horizon forecasts, especially when the number of time series observations is small. However, when the dynamic structure is unknown and for more complex dynamics and error structures such as the ones encountered in practice, one method stands out to have smaller forecast errors. This method forecasts the series of interest directly, rather than the common and idiosyncratic components separately, and it leaves the dynamics of the factors unspecified. By imposing fewer constraints, and having to estimate a smaller number of auxiliary parameters, the method appears to be less vulnerable to misspecification, leading to improved forecasts.

Determining the Number of Factors in Approximate Factor Models

In this paper we develop some statistical theory for factor models of large dimensions. The focus is the determination of the number of factors, which is an unresolved issue in the rapidly growing literature on multifactor models. We propose a panel C_p criterion and show that the number of factors can be consistently estimated using the criterion. The theory is developed under the framework of large cross-sections (N) and large time dimensions (T). No restriction is imposed on the relation between N and T. Simulations show that the proposed criterion yields almost precise estimates of the number of factors for configurations of the panel data encountered in practice.