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selection deterministic component

The objective of this book is the discussion and the practical illustration of techniques used in applied macroeconometrics. The plural here is as appropriate as ever in that the profession does not currently share a common view on the methodology of applied macroeconometrics. The different approaches are regarded as alternative , in fact it is very rare to see a combination of them applied by the same authors, it is also very difficult to see a combination of them published in the same journal, with the notable exception of the Journal of Applied Econometrics. Interestingly, up to the seventies there was consensus, both regarding the theoretical foundation and the empirical specification of macroeconometric modelling. The consensus was represented by the Cowles Commission approach which broke down in the seventies when it was discovered that this type of models ...did not represent the data, ... did not represent the theory... were ineffective for practical purposes of forecasting and policy... (Pesaran and Smith(1995)). The Cowles Commission approach was then substituted by a number prominent methods of empirical research: the LSE (London School of Economics) approach, the VAR approach, and the intertemporal optimization/Real Business Cycle approach. We shall discuss and illustrate the empirical research strategy of these three alternative approaches by interpreting them as different proposals to solve problems observed in the Cowles Commission approach. The presentation of each methodological approach is paired with extensive discussions and replications of the relevant empirical work. The bulk of empirical illustrations is related to the monetary transmission mechanism by considering benchmark data-set for the US and European economies. This choice allows us to have a common benchmark to address explicitly the differences in questions and answers provided by the different schools of thought.

Plan of the book

Our presentation is based on the conviction that the oldest concept in econometrics, namely identification, provides a natural unified framework to discuss the collapse of Cowles Commission models and the alternative strategies currently adopted in applied research. Therefore, in the first part of the book we introduce time series models and discuss extensively the importance of identification for time-series models. In the second part of the book we illustrate the Cowles Commission approach and the LSE, VAR, and intertemporal optimisation/calibration approaches providing applications.

Chapter 1 serves the purposes of giving a quick revision of basic econometrics and of introducing macroeconometrics. This is done analysing the economic problem of convergence and growth by using the data-set and replicating the results reported in Mankiw, Romer and Weil(MRW,1992). Traditional issues in econometrics, with particular emphasis on the mis-specification problem, are revised using the cross-section data set of the cited article. The importance of time series in econometrics is then stressed by illustrating the potential of the application of time series methodology to the MRW data-set.



Chapter 2 and 3 form the econometric basis for the illustration of all the different approaches to time-series. The discussion starts with the problems of temporal dependence of time series and its impact on the properties of estimator, the solution provided by asymptotic theory for stationary time-series is then discussed. Non-stationarity is introduced as the reason of the impossibility of using asymptotic theory to fix dependence. Cointegration is then considered as the solution in that it allows to specify a cointegrated VAR as the baseline stationary statistical model for non-stationary time-series. On such statistical model the fundamental problem of identification is discussed, with a separate treatment of identification of long-run equilibrium relationships and short-run simultaneous feedback. Within this common statistical framework, we introduce the criticisms to Cowles Commission Econometrics and our analysis of the alternative econometric methodologies currently used by the profession. Chapter 4 illustrates the Cowles Commission approach by considering specification, estimation and simulation of a simple IS-LM-AS-AD model fitted to US data. Chapter 5 illustrates the LSE methodology by introducing the diagnosis of the Cowles Commission problems provided within this approach as well as the proposed solution. Chapter 6, 7, and 8 apply the same strategy to VAR, Intertemporal optimisation GMM approach and calibration. The chapter on calibration would not exist in his actual format without the contribution of Marco Maffezzoli, who is jointly responsible for all this section. Marco and I have taught jointly the advanced econometrics option at the Bocconis Master in Economics, the book has greatly benefited from this experience, and not only the book.

Data, programmes and exercises

Data, programmes and exercises are available from the section of my homepage devoted to the book at the following address:

http: www.igier.uni-bocconi.it/personal/favero/homepage.htm

Applications presented in the book are performed by using different packages, such as E-VIEWs, PC-GIVE, PC-FIML, RATS and MATLAB. The data are provided in the appropriate format for the package used in the application and also in EXCEL format, to leave the reader free to experiment with the preferred software.

Exercises are not yet available at the date of publication of the book, but I plan to include them in the site as part of a project of continuous updating of the book. I really hope that the website will grow over time, also with contributions by the readers.

Acknowledgements

This book reflects my research and draws heavily from joint work with many co-authors. My grateful thanks go to Fabio-Cesare Bagliano, Rudi Dorn-busch, Francesco Franco, Francesco Giavazzi, David Hendry, Fabrizio Iacone, Jack Lucchetti, Alessandro Missale, Anton Muscatelli, Luca Papi, Hashem Pe-saran, Marco Pifferi, Riccardo Rovelli, Giorgio Primiceri, Sunil Sharma, Luigi Spaventa, Franco Spinelli, Guido Tabellini.



Over the years many colleagues have influenced and shaped my perception of how applied macroeconomics is and should be done. I am specially indebted to Larry Christiano, James Davidson, Roger Farmer, Carlo Giannini, Clive Granger, David Hendry, Soren Johansen, Katarina Juselius, John Muellbauer, Bahram Pesaran, Hashem Pesaran, Lucrezia Reichlin, Mike wickens and Harald Uhlig for having commented constructively on my work and having helped my understanding of different methodologies.

My work has benefited from comments by many students, especially those attending the Master in Economics (MEc) at Bocconi University, some of them have been also involved in writing, documenting, indexing the camera-ready copy of the book. Special thanks to Gani Aldashev, Marco Aiolfi, and Giorgio Primiceri for their help.

I firmly believe that the main motivation for my book is the promotion of cross-fertilization among different methodologies in applied macroeconometrics. A number of institutions have offered me the possibility of meeting and exchanging ideas with researchers from very different backgrounds. I am glad to acknowledge my debt to the Center For Economic Policy Research, to the Innocenzo Gasparini Institute for Economic Research and to Centro Interuniversitario di Econometria. I am also very grateful to Christian Zimmermann for his effort in constructing and developing the Quantitative Macroeconomics and Real Business Cycle home page:

http: ideas. uqam .ca/QMRBC/index.html,

a great website for getting in touch with research on applied macroeconometrics.

Colleagues at Queen Mary Westfield College, University of Ancona and Bocconi University have granted me over the years the possibility of working in friendly and open-minded environments. Special thanks to C.L.Gilbert, Jonathan Haskel, Chris Martin, Steve Pollock, Jack Lucchetti, Matteo Manera, Massimil-iano Marcellino and Bruno Sitzia.

Applied Macroeconometrics reflects the path of my research till 1999, and develops considerably ( I believe) on views and ideas expressed in a previous book in Italian entitled Econometria , edited by Nuova Italia Scientifica( now Carocci editore) in 1996. I am grateful to Gianluca Mori for his support in the old project, and for allowing me to develop freely the new project.

Financial support from MURST (Italian Ministry for University and Scientific Research) is gratefully acknoledged.



APPLIED MACROECONOMETRICS. AN INTRODUCTION

1.1 Introduction

Once upon a time there was consensus both on the theoretical foundations of macroeconomics and on the correct approach to macroeconometric modelling (see, for example, Pesaran-Smith [7]). Such consensus, which was built around the Cowles Commission approach to model building, broke down dramatically at the beginning of the seventies when it was discovered that ...the models did not represent the data...did not represent the theory... were ineffective for practical purposes of forecasting and policy... . The breakdown of consensus has been rather spectacular, but, as Faust and Whiteman ([2]) put it ... even more impressive are the deep rifts that have emerged over the proper way to tease empirical facts from macroeconomic data...

This book has the ambitious aim of discussing and illustrating the different approaches currently taken by the profession in doing applied macroeconomet-rics. We concentrate on the (large) subset of macroeconometrics dealing with time-series data. It is fair to say that the emergence of the deep rifts on the proper way to tease empirical facts from macroeconomic data has been paired with a deep awareness of the specificity of time-series data. We shall discuss the emergence of a plurality of approaches in macroeconomic modelling, within the framework provided by the statistical analysis of time series data. We begin our work with this introductory chapter, which reviews the basic in econometrics, describes the interaction between theory and data in applied work, and illustrates the importance of using time series instead of cross-section data in macroeconometrics.

1.2 From theory to data: the new-classical growth model.

Consider the Solow model of growth1 This model takes as given the saving rate s, the rate of growth of population n, while technology, A, grows at a constant rate g. There are two inputs: capital, K,and labour, L, paid their respective marginal productivity. Output, Y, is determined by a Cobb-Douglas function with constant returns to scale:

Yt = Kf{AtLtf-a 0<a<l (1.1)

The original reference is Solow ([9]). The data and the empirical analysis of this chapter replicate the results reported in Mankiw, Romer and Weil ([6]). For an excellent introduction to macroeconomic models of growth see Farmer ([!]) .



Lt=Lt-x (1+n)

(1.2)

At = At-i (1 + g)

(1.3)

Note that the number of effective unit of labour grows (approximately) at the rate (n + g). The model is built by considering the production function together with two accounting identities and an ad hoc relation between savings and output. The two accounting identities are:

where /, denotes investment, denotes savings and 6 represents the rate of depreciation of the capital stock K. (1.4) makes immediately clear that we are considering a closed economy with no government sector.

The relationship between output and saving is determined by assuming a constant marginal propensity to save s:

We define as к and у respectively the stock of capital per effective unit of labour (Kj AL) and the level of output per effective unit of labour {Yj AL) . By using all the equations in the model we have:

Equation (1.7) determines the pattern over time of the stock of capital per effective unit of labour. From this relation we can pin down the steady state value of k, by setting k* = kt+i for each i:

St=It

(1.4)


(1.5)


(1.7)


(1.8)

(1.8) makes clear that the steady state к is positively related to the saving rate and negatively related to the rate of growth of population, the rate of technological progress and the rate of depreciation of capital.



By substituting (1.8) in the production function and taking logarithms we can derive the per capita steady state output as :

In -M = \nA0+gt + --ln(s)---In (n+ <? + <$) (1.9)

\lt J 1 - al - a

(1.8) makes very specific predictions on the impact on output of the saving rate and the rate of depreciation of capital, the rate of technological progress and the rate of depreciation of capital.

It is natural at this stage to raise a question on the empirical support to such well specified predictions. Mankiw, Romer and Weil ([6]) choose to test the model on data from a cross-section of countries. Such data are available in a database constructed by Summers-Heston (1988), which contains series on real output, private and government consumption, investment and population for virtually all countries in the world, excluding planned economies. The data are available at annual frequencies. Mankiw,Romer and Weil concentrate on the variables of interest for the period 1960-1985. The rate of growth of population, n, is measured by the average rate of growth of population in working age (1564 years old). The rate of savings, s, is measured by the ratio of investment to GNP. n and s are averages for the period 1960-1985. у is measured by the log of GDP per working age person in 1985. (g + 6) is not directly observable and it is assumed constant at a value of 0.05. We concentrate on a sample of 75 countries labelled Intermediate by Mankiw, Romer and Weil and obtained considering non-oil producers countries with population higher than one million in 1960 and reliable data, thus excluding from the sample oil producers (as the bulk of GDP for such countries is not value added but extraction of existing resources), small countries and countries with low-quality data (receiving a grade of D from Summers and Heston). The data are contained, in EXCEL format, in the file MRW.XLS.

Now we have data and we have (1.9), which makes specific, theory-based predictions, on the relations between variables in our data set, the natural question is how we test empirically the Solow model?

The first point to note is that there is no stochastic structure in (1.9). Mankiw, Romer and Weil add a stochastic structure to the data by ignoring the difference between Y and Y* and by concentrating on the term A. In fact A reflects not only the state of technology but also other factors, such as natural resources, climate, institutions, therefore the following specification is adopted for A :

In Aq = a + £j

where a is a constant and £j represents a country-specific shock. (1.9) becomes now:



Inyi = In A0+gt + --ln(si)---Ы(щ +g + S) + Si (1.10)

1 - al - a

which forms the basis for the empirical investigation.

1.3 The estimation problem: Ordinary Least Squares

The basis for the empirical test of the predictions of the Solows growth model is the estimation of (1.10). Consider the estimation of the following model on our sample of 75 countries:

lnj/i = f30 + f31 In (si) + f32 In (щ + g + 8) +1

(1.11)

If the Solow model describes correctly the data, then the parameter (30 should capture the term InAo + gt, which is a constant of the cross-section of data, while /31 should be equal to and (32 should instead take the value of - j2-

Therefore, independent information on factor shares could be used to assess the magnitude of the estimated coefficient: Mankiw,Romer and Weil claim that data on factor shares suggest one-third as a plausible value for a and therefore the elasticities of yi with respect to S{ and (гц + <? + <*>) should be respectively 0.5 and -0.5. Moreover, under the null of the validity of the Solow model, we have a testable restrictions on the parameters, namely /31 = - (32-

To illustrate how estimation can be performed, consider the following general representation of our model:

у = X/3 + e

\VnJ

111 X\2

\XNi XN2

%Nk /

In our case N = 75, к = 3, the vector у contains 75 observations on per capita GDP while matrix X is (75 X 3). Note that the first column of X is made entirely of ones, the second column contains observations on



In (sj), while the third one contains observations on In (гц + g + 6). The vector /3 contains three parameters: a constant and the two elasticities of interest in our economic problem.

The simplest method to derive estimates of the parameters of interest is the Ordinary Least Squares (OLS) method. Such method chooses values for the unknown parameter to minimize, in some sense, the magnitude of the non-observable components. Define the following quantity:

e(/3) = y-X/3

where e (/3) is a (n X 1) vector . If we treat X/3, as a (conditional) prediction for y, then we can consider e (/3) as a forecasting error. The sum of the squared errors is then

S(/3) = e(/3)e(/3) OLS produces an estimator of /3, /3, defined as follows:

S(3) = mine 03)e 03)

Given /3, we can define an associated vector of residual ~e as e = у - X/3. The OLS estimator can be derived by considering the necessary and sufficient conditions for /3 to be a unique minimum for S :

i) Xe = 0

ii) rank(X) = к

Condition i) imposes orthogonality between the right-hand side variables on the OLS residuals, and ensures that the residual have an average of zero when a constant is included among the right-hand side variables (the regressors). Condition ii) requires that the columns of the X matrix are linearly independent: no variable in X can be expressed as a linear combination of the other variables in X.

From i) we can derive an expression for the OLS estimates:

Xe = X (y - X/3) = Xy - XX/3 = 0 3= (XX) 1Xy 1.3.1 Properties ofthe OLS estimates

We have derived the OLS estimator without any assumption on the statistical structure of the data. In fact the statistical structure of the data is not needed to derive the estimator but to define its properties. To illustrate such properties we refer to the basic concepts of mean and variance of vector variables.



Given a generic vector of variables, x

/ xA

\x j

we define the mean vector E (x) and the mean matrix of outer products E (xx) as follows:

E(x):

(E{Xl)\

\E(xn)J

E (xx) = E

/ x\ X\%2 .

x\ .

\XnX\ XnX2

xn )

( E (x\) E(x1x2) . . E(x1xn)\ E(x2xn)

Elxl)

\E(xnxx) E(xnx2) . . E(xl) J The variance-covariance matrix of x is the defined as follows:

var (x) = E (x-E (x)) E (x-E (x)) = = E (xx) - E (x) E (*.)

Note that the variance-covariance matrix is symmetric and positive definite, by construction. In fact, given an arbitrary A vector of dimension n, we have :

var (Ax) = Avar (x) A

The first relevant hypothesis for the derivation of the statistical properties of OLS regards the relationship between disturbances and regressors in the estimated equation. This hypothesis is constructed two parts: first it is assumed that



E (yi I х^) = x/3, this rules out the contemporaneous correlation between residuals and regressors (it is therefore valid if there are not omitted variables correlated with the regressors), second it is assumed that the components of the available sample are independently drawn. The second part of this assumption guarantees the equivalence between E (у, Xj) = x/3 and E(y, Xi, ...Xj, ...x ) = x/3. Using vector notation we have

E (у I X) = X/3 which can be written equivalently as

£?(eX)=0 (1.12)

Note that hypothesis (1.12) is very demanding. In fact, it implies that

E (e, I xi, ...хь ...x ) = 0 (г = 1, ...n)

The conditional mean is in general a non-linear function of (xi, ...Xj, ...x ), (1.12) requires that such function is a constant of zero. Note that (1.12) requires that each regressor is orthogonal not only to the error term associated to the same observation (E (xikSi) = 0 for all k) but also to the error tem associated to each other observations (E (xjkSi) = 0 for all j ф к). This statement is proofed by using the properties of conditional expectations.

Given that E (e X) = 0 implies, by the Law of Iterated Expectations, that E (e) = 0, we have

Е(ег J x:k)=E[E(£l J x) J xjk]=0 (1.13)

Then

E (eiXjk) =E[E (eiXjk \ xjk)] (1.14)

= E [xjkE (si I xjk)\ (1.15)

= 0 (1.16)

In the context of the Solow model (1.12) requires that s and n are independent from 6. Of course, such hypothesis is not going to hold in any time-series models when the time-series show some degree of persistence (in practice, always). Think of the simplest time-series model for a generic variable у :

yt=a0 + a1yt 1 +ut.

It is clear that if cii ф 0, then, although it is true that Щщ yt-i) = 0, E(wt i I yt-i) ф 0 and (1.12)is destroyed, without any omitted variable problem.



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