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2022 | OriginalPaper | Buchkapitel

10. Empirical Methods: Frequentist Estimation

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

The chapter starts with the Generalized Method of Moments estimator, describing its main properties, and applying it to the estimation of an equilibrium asset pricing model. After that, we explain the implementation of the Maximum Likelihood estimator. The Kalman filter, the main tool for the numerical evaluation of the likelihood on the state-space representation of the model, is discussed in detail. We estimate cyclical and trend components in US GDP and the unemployment rate. Finally, we compute the ML estimator to the Hansen (Journal of Monetary Economics 16:309–327, 1985) model of indivisible labor. We explain MATLAB programs provided to estimate structural parameters and generate some interesting properties of Growth models, as impulse responses to supply and demand shocks, and the decomposition of the variance of forecast errors.

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Fußnoten
1
Evidently, E t − 1[f0, y t)] = 0 and E t[f0, y t + 1)] = 0 are the same set of conditions.
 
2
This is a modified versión of “gmmestimation.m”, which was written by Cao Zhiguang (see https://​www.​mathworks.​com/​matlabcentral/​fileexchange/​12114-gmm).
 
3
The reader may want to change the estimation code to use as instruments a constant and lagged consumption growth.
 
4
It is well known that if the order of the dynamic system is higher than 1, it can be rewritten as a first order system by taking some lags as additional state variables.
 
5
For a more detailed description, see Hamilton [2], Chapter 13. We use the same notation as in that chapter.
 
6
The updating formula for a linear projection is based on the following: Let us assume that, conditional on the information set \( {\mathscr{Y}}_{t-1} \), the vector (Z 1, Z 2) follows a multivariate normal distribution: \( \left(\begin{array}{c}{Z}_1\\ {}{Z}_2\end{array}\right)\left|{\mathscr{Y}}_{t-1}\right.\sim N\left(\left(\begin{array}{c}{\mu}_1\\ {}{\mu}_2\end{array}\right),\left(\begin{array}{cc}{\Sigma}_{11}& {\Sigma}_{12}\\ {}{\Sigma}_{21}& {\Sigma}_{22}\end{array}\right)\right). \) Then, the probability distribution of Z 1 conditional on Z 2, as well as on the information set \( {\mathscr{Y}}_{t-1} \), is given by Z 1|Z 2, t − 1 ∼ N(μ 1 ∣ 2, Σ1 ∣ 2), where \( {\mu}_{1\mid 2}={\mu}_1+{\Sigma}_{12}{\Sigma}_{22}^{-1}\left({Z}_2-{\mu}_2\right) \), and \( {\Sigma}_{1\mid 2}={\Sigma}_{11}-{\Sigma}_{12}{\Sigma}_{22}^{-1}{\Sigma}_{21} \). Thus, denoting \( {Z}_1\equiv {\xi}_t,{Z}_2\equiv {y}_t-{\hat{y}}_{t\mid t-1}={H}^{\prime}\left({\xi}_t-{\hat{\xi}}_{t\mid t-1}\right)+{w}_t, \) we have, \( {\mu}_1\equiv {\hat{\xi}}_{t\mid t-1},{\mu}_2=0, \) Σ11 = P t ∣ t − 1, Σ22 = H P t ∣ t − 1 H + R, Σ12 = P t ∣ t − 1 H. Thus, \( {\mu}_{1\mid 2}={\hat{\xi}}_{t\mid t},{\Sigma}_{11\mid 2}={P}_{t\mid t}. \)
 
7
This definition applies here because we are analyzing an economy without endogenous growth.
 
8
As an exercise, the reader could estimate the parameters of the model using GDP and hours worked as observable variables.
 
9
The structure of these MATLAB files is inspired by those that Professor Peter Ireland has on his website (http://​irelandp.​com/​programs.​html). Of course, we encourage the reader to visit such a page to analyze the files that Professor Ireland has used for the frequentist estimation of different DSGE models in his research.
 
Literatur
1.
Zurück zum Zitat Clark, P. K. (1987). The cyclical component of U.S. economic activity. Quarterly Journal of Economics, 102, 797–814. CrossRef Clark, P. K. (1987). The cyclical component of U.S. economic activity. Quarterly Journal of Economics, 102, 797–814. CrossRef
2.
Zurück zum Zitat Hamilton, J. D. (1994). Time series analysis. Princeton University Press. CrossRef Hamilton, J. D. (1994). Time series analysis. Princeton University Press. CrossRef
3.
Zurück zum Zitat Hansen, G. D. (1985). Indivisible labour and the business cycle. Journal of Monetary Economics, 16, 309–327. CrossRef Hansen, G. D. (1985). Indivisible labour and the business cycle. Journal of Monetary Economics, 16, 309–327. CrossRef
4.
Zurück zum Zitat Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029–1054. CrossRef Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029–1054. CrossRef
5.
Zurück zum Zitat Hansen, L. P., & Singleton, K. (1982). Generalized instrumental variable estimation of nonlinear rational expectations models. Econometrica, 50(1982), 1269–1286. CrossRef Hansen, L. P., & Singleton, K. (1982). Generalized instrumental variable estimation of nonlinear rational expectations models. Econometrica, 50(1982), 1269–1286. CrossRef
6.
Zurück zum Zitat Newey, W. K., West, K. D. (1987). A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703–708. Newey, W. K., West, K. D. (1987). A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703–708.
Metadaten
Titel
Empirical Methods: Frequentist Estimation
verfasst von
Alfonso Novales
Esther Fernández
Jesús Ruiz
Copyright-Jahr
2022
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-63982-5_10