Università di Pavia 2007 Dynamic Regression Models Eduardo Rossi University of Pavia
Data Generating Process & Models Setup y t denote an (n 1) vector of economic variables generated at time t. The collection {y t, < t < } is called a (vector-valued) random sequence. An economic data set is a finite segment, {y 1,y 2,,y T }, of this infinite sequence. Data Generating Process (DGP): Joint probability law under which the sequence is generated, embodying all these influences. Assume that the data are continuously distributed. Eduardo Rossi c - Macroeconometria 2007 2
Data Generating Process & Models The DGP is completely represented by the conditional density D t (y t Y t 1 ) Y t 1 = σ (y t 1,y t 2,...) Y t 1 is the smallest σ field of events with respect to which the random variables y t j are measurable for all j 0. D t ( ) is allowed to depend on time, because the data are not assumed to be stationary and in particular allowance must be made for features such as seasonal variations, and changes in technology, regulatory regime, ecc. Eduardo Rossi c - Macroeconometria 2007 3
Models A dynamic econometric model is a family of functions of the data, of relatively simple form, devised by an investigator, which are intended to mimic aspects of the DGP. Model is a family of functions {M (y t,y t 1,...,d t ; θ),θ Θ}, Θ R p M D = Model of the complete DGP M E = Model of E (y t Y t 1 ) M V = Model of V ar (y t Y t 1 ) Models depend on a finite collection of parameters θ (p 1), Θ is the parameter space. Eduardo Rossi c - Macroeconometria 2007 4
Models DGP is not represented as depending on Θ. Parameterization is always a feature of the model, not explicitly of the DGP. Many different parameterization of DGP are possible. The axiom of correct specification is the assumption that there exists a model element that is identical to the corresponding function of the DGP. Eduardo Rossi c - Macroeconometria 2007 5
Models M D is correctly specified if there exists θ 0 Θ such that M D (y t,y t 1,...,d t ; θ 0 ) = D t (y t Y t 1 ) Similarly M E and M V are correct if the conditions M E (y t,y t 1,...,d t ; θ 0 ) = yd t (y t Y t 1 )dy M V (y t,y t 1,...,d t ; θ 0 ) = yy D t (y t Y t 1 )dy yd t (y t Y t 1 ) dy y D t (y t Y t 1 )dy The axiom of correct specification is implausible. Misspecification is common in practical modelling. Eduardo Rossi c - Macroeconometria 2007 6
Models Search for adequate approximation: i.e. M D, M E, M V are true in essentials. Specification tests that allow us to check whether a model fulfils the criteria set. Eduardo Rossi c - Macroeconometria 2007 7
Nonstochastic Time Variation The dependence of D t (y t Y t 1 ) on time can be through its stochastic arguments and/or through the variations in parameters: θ t. Define θ t = f (θ,d t ) The variables d t are determined outside the economic system. Dummy variables: Constructed by the investigator to represent a particular source of variation, rather than measured directly. Trend functions: t k, k 0; Seasonal dummies; Intervention dummies: 1 over certain periods, and 0 otherwise, representing particular policy regimes. Eduardo Rossi c - Macroeconometria 2007 8
The ARMADL model The application of dynamic modelling concepts to regression models. Let y t be a variable to be modelled and let z t (k 1) be a vector of explanatory variables, assumed weakly exogenous with respect to the parameters of interest. The conditioning set for this problem is F t = σ (z t,z t 1,...,y t 1, y t 2,...) the list of eligible of conditioning variables generally extends beyond those actually playing a role in the model. Eduardo Rossi c - Macroeconometria 2007 9
The ARMADL model Assuming linearity, the models of interest take the general form y t = δ d t + β 0z t + β 1z t 1 +... + β mz t m +α 1 y t 1 +... + α p y t p +u t + θ 1 u t 1 +... + θ q u t q where E (u t F t ) = 0. d t denotes dummy variables (intercept, seasonals, etc.). This is an AutoRegressive Moving Average Distributed Lag (ARMADL model). Eduardo Rossi c - Macroeconometria 2007 10
The ARMADL model The lagged innovations belong to the conditioning set F t. Using the lag operator where α (L) y t = δ d t + β (L) z t + θ (L) u t (1) α (L) = 1 α 1 L... α p L p θ (L) = 1 + θl +... + θ q L q β (L) = β 0 + β 1 L + β 2 L 2 +... + β m L m Eduardo Rossi c - Macroeconometria 2007 11
THE ARDL MODEL Explicitly omitting the MA component θ (L) yields α (L) y t = δ d t + β (L) z t + u t called AutoRegressive Distributed Lag Model, which can be estimated by OLS. This is probably the commonest type of model fitted in practice, just because of its semplicity. Eduardo Rossi c - Macroeconometria 2007 12
Common Factors Suppose there exists a lag polynomial ρ (L) such that α (L) = ρ (L) α + (L) β (L) = ρ (L)β + (L) and ρ (L) is a common factor in the lag structure of each variable. The common factor model has the general form of an ARDL model with an ARMA error term, α (L) y t = δ d t + β (L) z t + θ (L) u t ρ (L) α + (L) y t = δ d t + ρ (L) β + (L) z t + θ (L) u t u t WN ( 0, σ 2) α + (L) y t = δ + d + t + β + (L) z t + θ (L) ρ (L) u t (2) d + t represents the recombination of the dummies after transformation by 1/ρ (L). Eduardo Rossi c - Macroeconometria 2007 13
Common Factors Because of estimation difficulties, ARMA errors are not usually fitted in a regression model. In the context of the ARDL model (2) becomes an AR error term, α + (L) y t = δ + d + t + β + (L) z t + 1 ρ (L) u t This is a very popular scheme, and routines for computing the estimates have been widely implemented. However, the major reason for this popularity in the older literature was a fix for serial correlation in the residuals of static regressions rather a special form of dynamic model. Eduardo Rossi c - Macroeconometria 2007 14
Rational lags If α (L) = θ (L) in (1), the model can be put in the form y t = δ d t + β (L) α (L) z t + u t this is known as the rational lag model (Jorgenson, 1966). It is a way of rep resenting an infinite number of lags with a finite number of parameters, by having the coefficients decline geometrically beyond a certain point. The transformation back to α (L) y t = δ d t + β (L) z t + α (L) u t (3) is called Koyck transformation (Koyck, 1954). The rational lag model can be implemented in practice by estimating (3), but it is important not to neglect the induced moving average disturbance. Eduardo Rossi c - Macroeconometria 2007 15
Polynomial Distributed Lags Another scheme for reducing the number of parameters in a model with long lags (Almon, 1965). Consider for simplicity a model with just one explanatory variable y t = l β j x t j + u t j=0 where l is finite but large relative to the sample size. The problem is to make β j depend on a small number of underlying parameters in a plausible way, and a natural approach is to make the weights vary with j in a smooth manner. Let β j = α 0 + α 1 j + α 2 j 2 + + α p j p (4) where p l, and the α i, i = 0, 1,...,p are the parameters. Eduardo Rossi c - Macroeconometria 2007 16
Polynomial Distributed Lags Choosing p = 2 or 3 in (4) makes the β j lie along a quadratic or cubic curve. The constraints are imposed very simply by constructing new regressors z it = since then l j i x t j j=0 i = 0, 1,...,p l β j x t j = j=0 p α i z it. i=0 Eduardo Rossi c - Macroeconometria 2007 17
The Error Correction Model (ECM) The dynamics of a linear time series process can always be expressed in terms of the level and a lag polynomial in the differences. Given a polynomial α (z) of order p, there exists the equivalent representation α (z) = α (1) + α (z)(1 z) α (1) = α 0 + α 1 +... + α p where the coefficients of the p 1-order polynomial α (z) are p αj = α k j = 0, 1,...,p 1 k=j+1 Eduardo Rossi c - Macroeconometria 2007 18
The Error Correction Model (ECM) The first and second order cases are α 0 + α 1 z = (α 0 + α 1 ) α 1 (1 z) α 0 +α 1 z +α 2 z 2 = (α 0 + α 1 + α 2 ) (α 1 + α 2 )(1 z) α 2 z (1 z) This is the Beveridge-Nelson Decomposition. A further rearrangement yields where α (z) = α (1) z + α (z)(1 z) α (z) = α (z) + α (1) Eduardo Rossi c - Macroeconometria 2007 19
The Error Correction Model (ECM) Transforming β (z) similarly, the ARDL model can be written in the so-called Error Correction Model (ECM) form, α (z) y t = δ d t + β (L) z t α ( y t 1 θ z t 1 ) + ut where α = α (1) and θ = β (1) /α. This is only a reparameterization. The parameters θ can be thought of as representing the long-run equilibrium relations of the model, those that would prevail if z t were constant and u t = 0 for an indefinitely long period. Eduardo Rossi c - Macroeconometria 2007 20
The Error Correction Model (ECM) ARDL(1,1): y t = α 1 y t 1 + β 0 z t + β 1 z t 1 + u t α 1 < 1 y t = β 0 + β 1 L 1 α 1 L z t + 1 1 α 1 L u t y t = β 0 α j 1 z t j + β 1 α j 1 z t 1 j + j=0 y t = δ(l)z t + δ(l) = β 0 + β 1 L 1 α 1 L 1 1 α 1 L u t j=0 (1 α 1 L)δ(L) = β 0 + β 1 L α j 1 u t j j=0 Eduardo Rossi c - Macroeconometria 2007 21
Long run equilibrium δ 0 = β 0 δ 1 = (α 1 β 0 + β 1 ) δ j = α 1 δ j 1 j 2 Static equilibrium (all changes have ceased). We are treating (y t, z t ) as jointly stationary. The long-run values are given by the unconditional expectations: E (y t ) y = E (y t ) z = E (z t ). Eduardo Rossi c - Macroeconometria 2007 22
Long run equilibrium Since E (u t ) = 0 y = α 1 y + β 0 z + β 1 z y = (β 0 + β 1 ) (1 α 1 ) z = ( ) δ i z kz i=0 k long-run multiplier (total multiplier) of y with respect to z. Passing from z to z + 1, the new solution is ( ) δ(1)(z + 1) = y + δ i Impact multiplier δ 0 = β 0 i=0 instantaneous effect of an increase in z on y. Eduardo Rossi c - Macroeconometria 2007 23
Long run equilibrium Interim multipliers δ J = J δ i J = 0, 1, 2,... i=0 δ J = β 0(1 α 1 ) + (α 1 β 0 + β 1 )(1 α J 1) 1 α 1 J = 0, 1, 2,... the effect of a unit change in z after J periods. Standardized Interim Multiplier δ + J = δ J δ(1) δ + J = β 0(1 α 1 ) + (α 1 β 0 + β 1 )(1 α J 1) β 0 + β 1 portion of total adjustment that take place in the first J periods, J = 0, 1,.... Eduardo Rossi c - Macroeconometria 2007 24
Long run equilibrium When δ(1) 0 and 0 δ i 1 i = 1, 2,... The average lag is defined by µ = j=0 jδ j j=0 δ j Eduardo Rossi c - Macroeconometria 2007 25
Long run equilibrium ARDL(1,1): y t = α 1 y t 1 + β 0 z t + β 1 z t 1 + u t α 1 < 1 Nested models: Static regression α 1 = β 1 = 0 y t = β 0 z t + u t AR(1) β 0 = β 1 = 0 y t = α 1 y t 1 + u t Leading indicator α 1 = β 0 = 0 y t = β 1 z t 1 + u t Partial adjustment β 1 = 0 y t = α 1 y t 1 + β 0 z t + u t Distributed lags α 1 = 0 y t = β 0 z t + β 1 z t 1 + u t Common Factors β 1 = β 0 α 1 y t = +β 0 z t + ǫ t ǫ t = α 1 ǫ t 1 + u t First Difference α 1 = 1 β 1 + β 0 = 0 y t = β 0 z t + u t Dead Start β 0 = 0 y t = α 1 y t 1 + β 1 z t 1 + u t ECM with homogeneity constraint β 0 + β 1 + α 1 = 1 y t = β 0 z t + (α 1 1)(y t 1 z t 1 ) + u t Eduardo Rossi c - Macroeconometria 2007 26
Long run equilibrium Imposing the restriction α 1 + β 0 + β 1 = 1 1 α 1 = β 0 + β 1 y t y t 1 = α 1 y t 1 y t 1 +β 0 z t β 0 z t 1 + β 0 z t 1 +β 1 z t 1 +u t y t = y t 1 + α 1 y t 1 + (β 0 z t β 0 z t 1 ) + (β 0 z t 1 + β 1 z t 1 ) + u t Eduardo Rossi c - Macroeconometria 2007 27
Long run equilibrium y t = (1 α 1 ) y t 1 + β 0 z t + (β 0 + β 1 ) z t 1 + u t y t = (1 α 1 ) y t 1 + β 0 z t + (1 α 1 )z t 1 + u t y t = β 0 z t + (1 α 1 )(z t 1 y t 1 ) + u t Eduardo Rossi c - Macroeconometria 2007 28
Long run equilibrium In the ECM formulation, parameters describing the extent of short-run adjustment to disequilibrium are immediately provided by the regression. ECM terms are a way of capturing adjustments in a dependent variable which depend not on the levels of some explanatory variable, but on the extent to which an explanatory variable deviated from an equilibrium relationship, of the form y = θ x with the dependent variable. Eduardo Rossi c - Macroeconometria 2007 29
The Consumption function C t = AY t consume proportional to income. C t consumer target. A marginal propensity to consume (unobserved). We suppose A is observable, taking logs log C t = log A + log Y t c t = a + y t Eduardo Rossi c - Macroeconometria 2007 30
The Consumption function ECM c t = α c t + γ ( c t 1 c t 1 ) + ut the rate of growth of aggregate consumption as a function of target consumption. c t 1 target consumption at time t 1, c t 1 effective consumption at time t 1 c t = a + y t = y t c t = α y t + γ (a + y t 1 c t 1 ) + u t c t = aγ + α y t + γ (y t 1 c t 1 ) + u t Eduardo Rossi c - Macroeconometria 2007 31
The Consumption function Long-run equilibrium: E ( c t ) = aγ + γe (y t 1 c t 1 ) = 0 c = a + y C = AY Eduardo Rossi c - Macroeconometria 2007 32
Models of Dynamic Behavior Early econometric time series research: Fitting to time series static relationships (solutions of a comparative-static analysis, etc.). Unsatisfactory results. Lags matter, various approach to incorporating dynamic elements. Possible solutions: Early vogue for error dynamics (Common Factor Model). Restrictions involved are often found to be invalid. Add lags of the explanatory variables on a purely ad-hoc basis. Model systematic dynamics in terms of the costs of adjustment, the formation of subjective expectations, etc. Eduardo Rossi c - Macroeconometria 2007 33
The Partial Adjustment Model Desired level of an economic variable y t : y t = x tβ (e.g. optimal level of inventories). The partial adjustment model of planned end-of-period inventories is y p t = γy t 1 + (1 γ)y t, 0 γ < 1. At the start of the planning period (t) the firm aims at an inventory intermediate between the current actual and ideal levels. The regression equation is obtained by adding a disturbance term y t = y p t + u t = γy t 1 + (1 γ)x tβ Stability: γ < 1. If x t stationary so is y t. Eduardo Rossi c - Macroeconometria 2007 34
The Partial Adjustment Model If E[x t ] = µ x E[u t ] = 0 E[y t ] = γe[y t 1 ] + (1 γ)β µ x = β µ x since E[y t ] = E[y t 1 ] by stationarity. In a stationary steady state, the desired level is realized: y t = β x t. Unrealistic model. In a growing economy y p t lags behind a growing target even if the growth is smooth and predictable. Eduardo Rossi c - Macroeconometria 2007 35
The Partial Adjustment Model Cost function C = (y p t y t ) 2 + α(y p t y t 1 c y t ) 2 α > 0 c 0 quadratic in both the extent of departure from the optimum and the amount of adjustment required. c represents the conservatorism of agent s growth forecast. dc dy p t = 2(y p t y t ) + 2α(y p t y t 1 c y t ) 2(y p t y t ) + 2α(y p t y t 1 c y t ) = 0 y p t = y t α(y p t y t 1 c y t ) Eduardo Rossi c - Macroeconometria 2007 36
The Partial Adjustment Model (1 + α)y p t = y t + αy t 1 + αc y t y p t = 1 1 + α y t + α 1 + α y t 1 + α 1 + α c y t y p t = (1 γ)y t + γy t 1 + γc y t dove γ = α 1+α, 1 γ = 1 1+α. Substituting y t = y t 1 + y t. y p t = (1 γ)(y t 1 + y t ) + γy t 1 + γc y t Eduardo Rossi c - Macroeconometria 2007 37
The Partial Adjustment Model where y p t y t 1 = (1 γ)(y t 1 + y t ) + γy t 1 + γc y t y t 1 δ = (1 γ) µ = 1 + cα 1 + α = (1 γ)(y t 1 + y t ) (1 γ)y t 1 + γc y t = (1 γ)y t 1 (1 γ)y t 1 + (1 γ) y t + γc y t = (1 γ)(yt 1 y t 1 ) + [cγ + (1 γ)] yt [ ] 1 + cα = yt δ(y t 1 y 1 + α t 1) = µ y t δ(y t 1 y t 1) Eduardo Rossi c - Macroeconometria 2007 38
The Partial Adjustment Model The regression model is y t = y p t + u t y t = y p t y t 1 + u t = µ y t δ(y t 1 y t 1) + u t This model unifies several different approaches to modelling dynamics: a. ECM: y t = β x t y t = µβ x t δ(y t 1 β x t 1 ) + u t the coefficients of x t differ from those of x t 1 only by a factor of proportionality µ. Eduardo Rossi c - Macroeconometria 2007 39
The Partial Adjustment Model b. Partial Adjustment: c = 0 c. Common Factor Model: c = 1 then µ = 1, the rule is to adjust fully to shifts in the target: y t = µβ x t δ(y t 1 β x t 1 ) + u t y t = β x t (1 γ)(y t 1 β x t 1 ) + u t y t = β x t + γ(y t 1 β x t 1 ) + u t (1 γl)y t = (1 γl)β x t + u t if u t = 0 then y t = β x t = y t. y t = β x t + u t 1 γl Eduardo Rossi c - Macroeconometria 2007 40