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Introduction Most econometric relationships are subject to at least the following three problems Measurement Error The t

Estimation of Parameters in the Presence of Model misspecification and Measurement Error P. A. V. B. Swamy, George S. Tavlas, Stephen G. Hall And George Hondroyiannis. Introduction Most econometric relationships are subject to at least the following three problems Measurement Error

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Introduction Most econometric relationships are subject to at least the following three problems Measurement Error The t

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  1. Estimation of Parameters in the Presence of Model misspecification and Measurement ErrorP. A. V. B. Swamy, George S. Tavlas, Stephen G. HallAndGeorge Hondroyiannis

  2. Introduction • Most econometric relationships are subject to at least the following three problems • Measurement Error • The true functional form is unknown • Omitted variables • While there are specific techniques or procedures that deal with these problems one at a time there is no single approach which works for them all.

  3. This paper proposes a new estimation strategy which potentially deals with all forms of model misspecification simultaneously. Unlike some non-parametric techniques the proposed method is feasible with relatively small data sets It also offers a new interpretation of regression coefficients in the presence of misspecification and a formal representation of the biases which result

  4. 2. The Interpretation of Model Coefficients Consider the relationship between an endogenous variable and K-1 of its determinants …. Where in particular K-1 may be only a subset of the full set of determinants so that we have omitted variables . In addition we have measurement error = + , = + And we may be estimating the wrong functional form

  5. What is the econometrician trying to achieve? Our answer to this is to derive an estimate of the partial derivative of with respect to , and to test hypothesis about this. If you want to estimate the true model then there really is no alternative to specifying it correctly. However if you are only interested in partial effects then we offer another way forward.

  6. Consider the following time varying parameter model = + + … + All potential misspecification is captured in the time varying coefficients which offer a complete Explanation of y. Now the key question is what are the stochastic assumptions about the TVCs

  7. The correct stochastic assumptions about the TVC comes from an understanding of the misspecification which drives the time variation. Notation and assumptions Let denote the total number of determinants of y, this can not generally be known, but in general m>k-1. Now let and j=1…K-1 and g=k…m be the true coefficients on the underlying model, where the parameters vary because of either a non-linear functional form or truly changing parameters

  8. Now for g=k… let denote the intercept and j=1…K-1 denote the other coefficients of the regression of on …

  9. Then we can establish the following representation Theorem 1. = + + And = j=1…k-1 The first term is the true variation, the second the measurement effect, the third the omitted variables.

  10. And if we estimate a standard fixed coefficient regression model then the error term comprises all these effects. This means that the error term can not be independent of the included Xs (as it contains them). It is also impossible for valid instruments to exist in this example as if a variable is correlated with the included X variables it must be correlated with the errors (as the error contains the included Xs)

  11. Theorem 2: For j=1…K-1 the component of is the direct or ‘biased free’ effect of on with all the other determinants of held constant, and it is unique. The direct effect will be constant if the relationship between y and all the Xs is linear and time invariant. This is a useful interpretation of standard regression coefficients, which emphasises their potential biases.

  12. To make this approach useful as an estimation strategy we must have some way of identifying the bias part of the TVC. = + + Assumption 1Each coefficient of (1) is linearly related to certain drivers plus a random error,

  13. Assumption 2: For j=1,…,K-1, the set of P-1 coefficient drivers and the constant term divides into two disjoint subsets S1 and S2 so that + has the same pattern of time variation as and + has the same pattern of time variation as the sum of the last two terms on the RHS of (3) over the relevant estimation and forecasting periods. So we assume the drivers identify the bias component. This is like the dual of IV

  14. Assumption 3, The K-vector = Of errors in (4) follows the stochastic equation = + Assumption 4, The regressor of (1) is conditionally independent of its coefficient given the coefficient drivers for all j and t

  15. A vector formulation of the model Where And = Then or

  16. Theorem 3 Under Assumptions 1-4 E = And Var = Where is the covariance matrix of

  17. 3. Identification and Consistent Estimation of TVC The fixed coefficient vector is identified if . has full column rank. A necessary condition for this is that T>Kp. The errors are not identified Thus assumptions 1-4 make all the fixed parameters of the model identifiable. This does not happen if we assume random walk TVCs

  18. Estimation under assumptions 1-4 IV estimation works under very extreme assumptions, we argue that in the presence of general mis-specification IV can not work

  19. Estimation under 1-4 Why IV can not work From Theorem 1 = + + And = j=1…k-1 And so if we estimate fixed parameters the second and third term go into the error. An instrument can not be correlated with x and uncorrelated with the error.

  20. Practical Estimation The TVCModel can be estimated by an iteratively rescaled generalized least squares (IRSGLS) method developed in Chang, Swamy, Hallahan and Tavlas (2000). Or alternatively it can be specified in state space form and estimated by maximum likelihood.

  21. Choice of coefficient drivers At present this is somewhat arbitrary as in the case of selecting instrumental variables. However some guidelines… Coefficient drivers should Be correlated with omitted variables or structural change Be correlated with any policy change

  22. Selecting a subset of the drivers The bias can be removed by selecting a subset of the drivers and removing their effect. How do we choose this subset (S1 and S2): Theory often suggests a sign or value for the true parameter. Coefficient drivers which give coefficients outside this range should be rejected Beyond that we have the following sugestion

  23. 3.6 Bayesian averaging of coefficient drivers. Each possible set or subset of drivers may be seen as a different model of the bias component. If we assign a prior probability to each admissible model then we can compute the prior density for each M = And then average over these =

  24. By Bayes Theorem the Posterior probability of M is = And the posterior for is =

  25. A Bayes estimator of is then = +

  26. Conclusions All models are subject to a range of sources of bias We offer a new representation of this bias This implies a potentially powerful way of estimating consistent effects It also offers a fundamental criticism of instrumental variables in practise

  27. An Example

  28. A PORTFOLIO BALANCE APPROACH TO EURO-AREA MONEY DEMAND IN A TIME-VARYING ENVIRONMENT Stephen G.Hall Leicester University, Bank of Greece and NIESR George Hondroyiannis Bank of Greece P.A.V.B. Swamy U.S. Bureau of Labor Statistics George S. Tavlas Bank of Greece

  29. 1. Introduction Money has played an important role in the setting of monetary policy within the Euro area since the inception of the ECB A reference value of 4.5% growth in M3 has been set but rarely achieved This has raised concerns about • The possibility of a monetary overhang • The reference path itself • The stability of the money demand function

  30. Between 2001 and 2003 a number of attempts were made to repair the money demand function but these all proved unstable beyond 2003 (Beyer, Fischer, and von Landesberger, 2007; Fischer, Lenza, Pill, and Reichlin, 2007) This paper considers money demand from a portfolio balance perspective and emphasizes the role of wealth. Unfortunately there are not good data for euro wide wealth. Our view is that there is a stable but complex money demand function which really requires much better wealth data to uncover

  31. In the absence of this good data we propose to use two techniques to support our contention A structural VEC approach augmented by a range of effects to proxy what we believe has been happening to wealth A TVC approach which is designed to reveal structural underlying parameters which are being obscured by a range of econometric problems including omitted variables, measurement error and misspecified functional form

  32. 2. Theoretical and empirical underpinnings The basic portfolio theory implies an equilibrium relationship of the form Assuming rate of return homogeneity And assuming log linearity and reparameterising gives

  33. 2.3 Existing models The current ‘workhorse’ model used by the ECB has the following long run This is known to be unstable since 2001 but the model currently in use still uses this long run as no stable alternative has been found The income elasticity of 1.3 suggests omitted wealth effects with wealth growing faster than income. Re-estimating this model using ECB data to 2006:3 gives the following results

  34. (m-p), y, , (m-p), y, (m-p), y, , (m-p), y, Which clearly shows no cointegration We can see why in terms of velocity in the next picture

  35. 3. Data and Empirical Results So given the lack of good data on wealth and its components (especially perhaps house prices) we construct a set of proxy variables. A stock market index An HP filtered version of the stock market index A split trend starting in 2001 (ST1) A complex dummy capturing stock market effects between 1988 and 1994 (ST2) (German unification, the hard EMS etc) The change in oil prices was also used in the VAR to help with the explanation of inflation

  36. We then apply 9 identifying restrictions to allow us to give a structural interpretation to the vectors for money, income and the wealth to income ratio. Here is the money demand CV

  37. The Dynamic Money demand function

  38. Recursive estimates of the long run coefficients

  39. Stability tests

  40. CUSUM and CUSUMSQ tests

  41. So a highly stable money demand function once we have allowed for these other factors which amount to wealth proxies.

  42. 3.2 TVC results Next we present the results for the TVC estimation. We investigate the following model Where Which implies the long run is our primary interest

  43. The coefficient drivers are all the variables making up the dynamic VEC money demand function. The equation for the coefficient allows us to break up the evolution in the coefficients into that caused by simultaneous equation bias or omitted variable bias and then remove this.

  44. This is the average effect, the next three slides show the time variation in the total coefficient and the bias free coefficient • The interesting thing here is that in each case the bias free coefficients are remarkably stable

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