Consistent estimation of asset pricing models using generalized spectral estimator

Consistent estimation of asset pricing models using generalized spectral estimator Jinho Choi Indiana University April 0, 009 Work in progress Abstract This paper essentially extends the generalized spectral estimation of Berkowitz (00) to provide a consistent generalized spectral estimator (GSE), considering all the information available, possibly with in nite dimensions, based upon Escanciano (006). Our estimator can entertain the strengths of the Berkowitz-GSE over the standard GMM. In contrast, more importantly, the newly proposed estimator has consistency which the Berkowitz-GSE is de cient in, overcoming Domínguez and Lobato s (004) critique on the identi ability of the GMM approach. Furthermore, our estimator is more general, based upon fairly relaxed assumptions for its asymptotic behaviors, than the Berkowitz-GSE. Finally, as an empirical application, using the proposed estimation strategy, we estimate the standard consumption-based asset pricing model in Hansen and Singleton (98) to investigate the possibility that the equity premium puzzle may be due to underidenti cation of risk aversion parameters. JEL Classi cation: C3; C Keywords: Identi cation; Conditional moment; Spectral analysis Department of Economics, 05 Wylie Hall, 00 S. Woodlawn, Bloomington, IN 47405, U.S.A.; choi5@indiana.edu. I would like to thank my advisor, Professor Juan Carlos Escanciano for many helpful comments and suggestions. All errors are mine.

Introduction Since introduced by Hansen (98), the generalized method of moments (GMM) has been widely used to estimate conditional moment restrictions implied by economic theories. Among a variety of the advantages, GMM has immediately gained popularity in econometrics mainly because no distributional assumptions are needed. For instance, as micro-founded macroeconomics becomes standard in the decade, a great number of literature in nance and macroeconomics has employed the GMM approach to estimate a speci c type of conditional moment restrictions called the Euler equations, which characterize the agents decisionmaking resulting from the utility maximization; for illustration, see Hansen and Singleton (98), Harvey (99), and Gali and Gertler (999). Although most applications of GMM are based upon the time domain approach, Berkowitz s (00) work is remarkable in the sense that he proposes a frequency domain version of GMM, named generalized spectral estimator (GSE). In estimating parameters of interest, both approaches convert conditional moment restrictions into unconditional moments, whereas subsequent procedures would be entirely di erent across the two approaches. To clarify this point, suppose we derive arbitrary conditional moment restrictions from the theory as follows: E[h(Y t ; 0 )jx t ] = 0 a:s: for a unique value 0, where R p () Then, to apply the standard GMM or GSE appoach, econometricians take into account unconditional moments as their moment conditions: E[h(Y t ; 0 )g(x t )] = 0 a:s: for any given g() () Given the population condition (), the standard estimation strategy in the literature is rmly based upon the assumption that 0 is globally identi ed, arbitrarily selecting a nite number of unconditional moments out of in nite candidates of g(x t ), and then minimizing a sample analogue of the objective function to yield GMM-type estimators. However, Domínguez and Lobato (004) point out that the key assumption in the GMM literature may be seriously awed and thus give rises to nontrivial problems in terms of consistency because the unconditional moments utilize fairly limited information on the data generating process, In his earlier working paper version, Berkowitz (996) named the proposed methodology as Spectral GMM. However, Chacko and Viceira (003) also use the term to denote their estimation strategy for continuous-time stochastic models based upon the characteristic function. In order to avoid confusion, we do not use the term Spectral GMM in this paper.

showing that their minimum distance estimator (DL) outperforms the GMM estimator in identifying parameters across di erent types of data generating process. Considering this possibility of underidenti cation may have substantial implications on the empirical literature relying on the GMM-type estimation, providing some clues to solve several interesting problems, including the equity premium puzzle. This paper basically extends Berkowitz (00) to propose a generalized spectral estimator, possibly with a in nite dimension, based upon Escanciano (006), which employs a generalized spectral distribution to provide goodness-of- t tests for the parametric conditional mean. Our estimator can entertain the strengths of the Berkowitz GSE over the traditional GMM (e.g., focusing on a subset of frequencies, no need to consider the weighting matrix). In contrast, from the perspective of Domínguez and Lobato (004), the proposed estimator is consistent whereas the Berkowitz s GSE may not be consistent as a result of lack of identi cation. In this sense, our estimator can be considered as spectral-dl. As a simple application, we estimate the classical consumption-based asset pricing model in Hansen and Singleton (98) to compare our estimation strategy with the two existing methods: the Berkowitz s GSE, and the standard GMM. The paper proceeds as follows. Section overviews identi cation issues in the GMMtype estimation, illustrating with an example in Domínguez and Lobato (004). Section 3 proposes an alternative generalized spectral estimator to Berkowitz s (00) GSE and provides the asymptotic theory. Section 4 presents the estimation and testing results for a consumption-based asset pricing model. Section 5 concludes. Identi cation issues in GMM With the rational expectation prevailing in several elds of economic theory, conditional moment restrictions are widely being used to describe model equilibrium, in which researchers are eventually interested. In the literature, the most popular estimation strategy for conditional moment restrictions has been generalized method of moments (GMM) proposed by Hansen (98). However, despite several advantages, to implement GMM in practice, nding appropriate instruments with relevance and validity is fairly challenging. In this regard, there is a growing number of literature raising a variety of questions about the identi ability of GMM. For instance, a line of literature actively examines on namely weak identi cation problem, caused by faint relevances between instruments and endogenous variables. See Stock and Wright (000), Stock et al. (00), Andrews and Stock (005).

Furthermore, more recent work suspects that the GMM approach may even fail to identify parameters especially when a model is de ned by conditional moment conditions. Domínguez and Lobato (004) show that the GMM s key assumption of global identi cation may be seriously awed, providing a consistent estimator. In addition, Hsu and Kuan (008) use Fourier-coe cient based to propose a consistent estimator, which is favorably compared with Domínguez and Lobato s. In this section we use Domínguez and Lobato s (004) simple illustration to explore potential identi cation failures of the GMM-type estimation. Consider a univariate random variable Y with the conditional mean of E(Y jx) = 0X + 0 X. Assume the true value of 0 equals 5=4 and V (Y jx) is constant. Furthermore, suppose that an econometrician correctly speci es the model and chooses the optimal instrument W = X + X. Then, he can construct the unconditional moment condition: E[(Y X X )W ] = E[(Y X X )(X + X )] (3) = E[(E[Y jx] X X )(X + X )] = E[( 0 )fx 4 + ( 0 + 3)X 3 + ( 0 + )X g] = 0 Then, when the conditioning variable X follows an N(0; ), the last equality in the condition (3) holds only if = 0 = +5=4, which implies identi cation. In contrast, when X follows an N(; ), either = 3 or 5=4 as well as the true value of +5=4 makes the unconditional moment equal to zero, showing no identi cation or underidenti cation. This simple example manifests the case that the global identi cation assumption in GMM may not hold: E[h(Y t ; )g(x t )] = 0 a:s: for some g(x) ; = 0 (4) Intuitively, we can interpret the case (4) against the identi cation assumption in GMM as follows. For any given conditional moment restriction (), one can generate an in nite number of unconditional moment restrictions (or instruments g(x)) in (). However, in practice, selecting only a few instruments may lead to inconsistent estimation because replacing conditional moments by unconditional moments may require losing crucial information from the original restrictions. To overcome the risk of potential underidenti cation, Domínguez and Lobato also propose an alternative estimator using the whole information about 0 in the conditional moment restriction (). Using Theorem 6.0 (iii) in Billingsley (995), one 3

can obtain the following equivalence: E[h(Y t ; 0 )jx t ] = 0 a:s:, H( 0 ; x) = 0 for almost all x R d (5) where H(; x) = E[h(Y t ; )I(X t x)] and I() indicator function. Then, the population parameter 0 can be recovered by minimizing the measure of the distance of H(; x) from 0, 0 = arg min H(; x) dp Xt (x) (6) where P Xt is the probability density function of the random vector X t. Therefore, corresponding to (6), Domínguez and Lobato propose a minimum distance estimator (DL), ^DL = arg min n 3, h(y t ; )I(X t X l ) (7) l= which is consistent and asymptotically normal. Furthermore, using a simulation study (Table I, p. 608), they show that in terms of bias, standard error and mean square error, the DL estimator outperforms the GMM estimator for either X s N(0; ) or X s N(; ), which we analytically considered above. With this background, the following section discusses spectral estimators in the frequency domain framework, corresponding to the standard GMM. 3 Generalized spectral estimation 3. Generalized spectral estimators (GSE) Given that the time domain framework is dominant in econometric analysis, why should we still need to pay attention to the frequency domain approach as considered in this paper? It is because some di cult problems under one framework may be easily resolved using the other. Furthermore, in general, using the frequency domain allows us to assess the contributions of individual frequencies to overall identi cation, as well as to reduce computational burden relative to the time domain approach. Motivated by Durlauf (99), Berkowitz (00) proposes a generalized spectral estimator in the frequency domain, corresponding to standard GMM estimators in the time domain approach. Under his framework, selecting lags of the Euler residual, h(y t j ; ) as instruments 4

replaces the conditional moment restriction () with E[h(Y t ; 0 )h(y t j ; 0 )] = 0 a:s: for j, (8) implying that the autocovariance function (j) = 0 for j and thus making the associated spectral density f ht( 0 )(u) = X j= (j)e iju = 0 (0), (9) where u denotes frequency and h t () h(y t ; ). From (9), we can observe that at = 0, the spectral density of h(y t ; 0 ) is at, i.e., f ht( 0 )(u) = = over its entire support, otherwise deviating from the constant value. spectral density from the constant as follows: S() = 0 Using this fact, one can formulate the distance of a fht (u) du; (0; ) (0) Then, using the usual Cramér-von Mises (CvM) norm to measure the distance, Berkowitz proposes a generalized spectral density estimator (BGSE). b BGSE = arg min bs() d where S() b = 0 0 fht b (u) b du () From the viewpoint of identi cation, however, BGSE cannot avoid the Domínguez and Lobato s critique because the condition (8) still assumes that the autocovariance function yields zero if and only if = 0. In what follows, we will show that this identi cation assumption is not always valid, thus highlighting the potential absence of identi cation in the context of Berkowitz (00). Let us consider a simple linear process. Y t = 0 X t + " t with E[" t jf t ] = 0; () where F t is the - eld generated by the conditioning set I t = (X t ; Y t ; X t ; Y t ; :::) 0. Then, assuming the model is correctly speci ed, we obtain E[Y t jf t ] = 0 X t, and " t () = Y t 0 X t. Given the condition, let us check if the Berkowitz s identi cation assumption (8) is valid, i.e. E[" t ()" t j ()] = 0 a:s: for j () = 0 (3) 5

Hence, we can rewrite the autocovariance function as follows: E[" t ()" t j ()] = E[(Y t X t )(Y t j X t j )] = E[(E(Y t jf t ) X t )(Y t j X t j )] = ( 0 ) E[X t X t j ] + ( 0 )E[X t " t j ] (4) Then, identi cation may fail because the equality in (4) holds when for all j ; = 0 or = 0 + E[X t" t j ] E[X t X t j ] if E[X t X t j ] 6= 0 (5) Speci cally, for an AR() process (i.e.,x t = Y t ), we can easily show that the autocovariance function (4) equals to zero when = = 0 as well as the true 0. Therefore, we can rewrite (3) as E[" t ()" t j ()] = 0 a:s: for j () = 0 or = 0 which implies that identi cation may fail. In order to verify the possibility of underidenti cation, we simulate AR() process y t = 0:8y t +" t ; " t s N(0; ). To minimize the dependence upon the selection of initial values y 0, we generate N observations and then wash out the initial N 0. Figure presents the objective function of squared sample autocovariance function E n [" t ()" t j ()] along the grid of possible values for [0:5; :5], setting with N = 000; N 0 = 000; j = 5. As the gure shows, we can verify that the objective function is minimized at zero when = :5 (= =0:8), as well as the true value of 0:8. Accordingly, to exclude the possibility that parameters of interest may not be identi ed, we extend Berkowitz (00) to propose a consistent estimator considering all the information available, possibly with in nite dimensions, based upon the The two terms associated with the second solution in (5) can be obtained as follows: X E[X t " t j ] = E[Y t " t j ] = E[( h 0" t h )" t j ] h=0 = j 0 ; for j 0 j < (6) X X E[X t X t j ] = E[Y t Y t j ] = E[( h 0" t h )( k 0" t k j )] = X h=0 k=0 h=0 k=0 X j h 0 k 0 0Cov(" t h ; " t k j ) = 0 (7) 6

Figure : Underidenti cation in AR() with 0 = 0:8 testing methodology by Escanciano (006). Escanciano (006) introduces the use of a generalized spectral distribution for testing martingale di erence hypothesis (). Among several advantages, using the spectral distribution allows us to skip the choice of any kernel and bandwidth for testing. Moreover, unlike Berkowitz (00), we can escape the potential identi cation problem if converting the test statistics proposed by Escanciano (006) into minimization criteria. To obtain a consistent estimator, we follow the notations and procedure to derive the integrated generalized spectral tests in Escanciano (006). Let f(y t ; X 0 t )g t be a strictly stationary and ergodic time series process de ned on the probability space (; F; P ), where Y t R dependent variable and t = (Y t ; X 0 t ) 0 R m ; m N, is the explanatory random vector including the lags of Y t and X t. Furthermore, we denote the conditioning set at time t as I t = ( 0 t ; 0 t ; :::) 0. Then, let us consider a parameterized conditional 7

moment restriction implied from an economic theory: E[h(Y t ; 0 )ji t ] = 0 a:s: for a unique 0 R p, (8) which is equivalent to E[h(Y t ; 0 )j t j ] = 0 a:s: 8 j, for a unique 0 R p. (9) Then, by selecting an appropriate function from the family of functions F = fw(; x) : x R s g satisfying Lemma in Escanciano (006), we can rewrite the restriction (9) using a generalized measure of dependence j;w () as j;w (x; 0 ) = E[h(Y t ; 0 )w( t j ; x)] = 0 a:e: in R s, s N, j. (0) While among popular examples of w(; x) are the exponential functions or indicator functions, we maintain the general notation in this derivation. For a list of the literature using di erent weighting functions, see Escanciano and Velasco (006). Then, applying the Fourier transform to the functions j;w (x; ) j= spectral density where i = p f w (u; x; ) = X j= ; and u denotes frequency. j;w (x; )e iju ; 8u [ ; ]; x, we obtain a Following Escanciano (006), we construct a generalized spectral distribution function as H w (; x; ) = 0 = 0;w (x; ) + f w (u; x; )du; (0; ) () X j= sin j j;w (x; ) j As with (9), a careful investigation reveals that evaluated at = 0, the spectral distribution function yields a constant value over the entire support: H w (; x; 0 ) = 0;w (x; 0 ) () Then, let us consider a sample {Y t ; b I t } n where b I t = ( 0 t ; 0 t ; :::; 0 0) 0, and denote 8

the sample conditional moment restriction as h t () b h t (Y t ; ). Then the sample analogue of () becomes bh w (; x; ) = b 0;w (x; ) + j= = sin j b j;w (x; )(n j =n) j (3) P where b j;w (x; ) = n n j t=j h t()w( t j ; x) for j, n j = n j +, and (n j =n) = a nite-sample correction factor. In the spirit of Berkowitz (00), we can formulate the deviation of the sample spectral distribution H b w (; x; )=b 0;w (x; ) from the constant = H w (; x; 0 )= 0;w (x; 0 ), or equivalently the distance between H b w (; x; ) and H b 0;w (; x; ) = b 0;w (x; ): n = n S n;w (; x; ) = bhw (; x; ) H0;w b (; x; )o p sin j = n = j b j;w (x; ) j j= = p h t ()q t;w (; x; ), n (4) where q t;w (; x; ) P t j= (n=n j) = p sin j j w( t j ; x). Therefore, using the Cramér-von Mises (CvM) norm, we can measure the distance S n;w () as Dn;w() = js n;w (; x; )j W (dx)d (5) where = [0; ] and W () is an integrating function associated with the weight family F de ned above. Given the last expression for S n;w (; x; ) in (4), the CvM norm can be considered as application of the Integrated Conditional Moment (ICM) statistic proposed by Bierens (98). Accordingly, it follows from the minimization of the norm (5) that we obtain a generalized spectral estimator (CGSE) as b CGSE = arg min D n;w() (6) Speci cally, if we choose the exponential function for w( t j ; x) in the dependence measure (0), (i.e., w( t j ; x) = exp(ix 0 t j ); x R m ) with a selection of the cumulative distribution function of a standard normal random variable for the integrating function in W (), then 9

(6) can be rewritten as b CGSE = arg min = arg min = arg min D n;c() js n;c (; x; )j d(x)d (7) j= b 0;C n j (j) t=j h t ()h s () expf ( t j s j ) g Note that given the exponential weighting function, we can easily obtain moment conditions associated with BGSE by di erentiating characteristic functions. s=j In this sense, we nd that CGSE can be considered as a generalized version of BGSE. Furthermore, the proposed estimator attains useful asymptotic behaviors such as consistency and asymptotic normality as provided in the next subsection. 3. Asymptotic theory 3.. Consistency Assumption The parametric space is compact in R p. The true parameter 0 belongs to the interior of. Assumption fy t ; t g t is a strictly stationary and ergodic process. Assumption 3 h(y t ; ) is continuous at each with probability one and satis es E[sup jh(y t ; )j] <. Assumption 4 E[h(Y t ; )jx t ] = 0 a:s: if and only if = 0. Theorem Let Assumptions -4 hold. Then b CGSE! a:s: 0 : Proof. Due to (), the population objective function DC () is uniquely minimized at 0. Furthermore, it follows from Assumptions -4 and stanardard M-estimator theory that the sample analogue Dn;C () converges uniformly in probability to D C (). Then, by Amemiya (985, Theorem 4..), it completes the proof. 0

3.. Asymptotic normality Denote = (; x 0 ) 0 and consider the process S n () = p n X n h t()q t () where q t () = q t;c (; x; ). Then under standard regularity conditions, similarly to Bierens (990), and p n( b 0 ) = D( 0 ) p n h t ( 0 ) @ @ 0 h t( 0 ) + o p () D() = E[( @ @ 0 h t())( @ @ 0 h t()) 0 ]; b(; ) = E[ @ @ 0 h t()q t ()]. Hence, by Lemma 3 in Bierens (990), we can rewrite S n () as S n () = p n h t ()q t () = p n where t () = q t () + b( 0 ; ) 0 D( 0 ) @ @ 0 h t ( 0 ). h t ( 0 ) t () + o p (); Assumption 5 h(y t ; ) is once continuously di erentiable in a neighborhood of 0, satisfying E[sup 0 j _ h(y t ; )j] < where 0 is a neighborhood of 0 and _ h(y t ; ) = @ @ h(y t; ). Assumption 6 h(y t ; 0 ) is a martingale di erence sequence with respect to f s ; s tg. Assumption 7 E[h 4 (Y t ; 0 ) kx t k + ] < and the density of the conditioning variables given the history is continuous and bounded. Theorem Let Assumptions -7 hold. Then p n( b CGSE 0 )! d N(0; ) where = _H H _ 0 d _H( ) H _ 0 ( ) ( ; )d( )d( ) _H H _ 0 d with _ H() = E[ _ h t ( 0 )q t ()] and covariance matrix ( ; ) = p lim n! n P n h t t ( ) t ( ): Proof. Let us denote S n () = S n;c (; x; ). Then the minimization of the objective function in (7)

yields the following rst order conditions: _S n ( b )S n ( b )d(x)d = 0 (8) Then, by the mean value theorem, we obtain _S n ( b )S n ( 0 )d(x)d + _S n ( b ) S : n ()d(x)d ( b 0 ) = 0 where = 0 + ( order conditions as ) b for some random [0; ]. Therefore, we can rewrite the rst p n( b 0 ) = = " " _S n ( b ) S _ p n ()d(x)d n ( n ( n ) ( h_ t ( b )q t () n ) ( h_ t ( b )q t () _S n ( b )S n ( 0 )d(x)d ) h_ t ()q t () d(x)d# p n ) # h t ( 0 )q t () d(x)d. Using the continuous mapping theorem, combined with Assumption 5, Lemma and below completes the proof. Lemma Let be a consistent estimator of 0, and Assumptions -7 hold. Then n h_ t ( )q t ()! a:s: _H() = E[ _ h t ( 0 )q t ()] uniformly in. Proof. Let us denote _ H n () = n H _ n ( ) _H( 0 ) X n j= h_ t ()q t () and H() _ = E[ h _ t ()q t ()]. Then consider H _ n ( ) _H( ) sup H _ n () + H( _ ) _H( 0 ) _H() + H( _ ) _H( 0 )! a:s: 0, which is implied by the uniform law of large numbers, the consistency of and the continuous mapping theorem under Assumptions -7.

Lemma Let Assumptions -7 hold. Then S n ( 0 ) = p n h t ( 0 )q t () ) S where ) denotes weak convergence in C [] and S is a Gaussian process on, with zero mean and covariance function ( ; ). Proof. Let us denote S n () as S n. Then, by the Prohorov s Theorem, it su ces to show that the nite-dimensional distributions ( dis) of the random function S n converges to normal distribution and that S n is asymptotically tight. The rst part of the proof can be easily obtained by applying a version of martingale di erence central limit theorem. See Bierens (994, Theorem 6..7). To prove the tightness of S n, we de ne n () as tight random functions on such that P [S n = n ] " for an arbitrary " and need to show that n is tight. For the tightness of n, according to by the Kolmogorov-Cencov criterion, we need to show that there exists a constant C such that E j n ( 0 )j C and E j n ( ) n ( )j C k k k+ for 8 0 ; ; ; 9; > 0;and k is the dimension of. Following Bierens and Ploberger (997), de ne the stopping time (M) = supft nja t ( 0 ) nm; B t nmg for an arbitrary 0 and M > 0, with A t () = P t j= h j j () ; B t = P t j= h jk j. Then, using Burkholder s inequality to n proves the rst condition of the criterion when = k + : E j n ( 0 )j k+ C k+ (=n k+ )E( P (M) h t t ( 0 ) ) k+ C k+ M k+. For the proof of the second part, by applying Burkholder s inequality, the Lipschitz condition and the de nition of the stopping time (M), we can obtain E j n ( ) n ( )j k+ P = (=n k+ )E (M) h t ( t ( ) t ( )) k+ P(M) k+ C k (=n k+ )E h t ( t ( ) t ( )) P(M) k+ C k (=n k+ )E h t Kt k k k+ C k k k k+ M k+ : 3

which implies that n is tight, and therefore S n converges weakly to a Gaussian process S. For details, see Bierens and Ploberger (997) 4 Application: consumption-based CAPM (In progress) It is widely recognized that taking data to standard consumption-based asset pricing models using GMM generates too high risk aversion ( Equity premium puzzle ). Among a huge number of alternative models, models with habit persistence (Campbell and Cochrane (999)) or long-run risk factor (Bansal and Yaron (004)) have drawn much attentions from researchers. However, Berkowitz (00) proposes such a high level of risk aversion may be due to the noise in the high frequency data, using the generalized spectral estimation technique. We employ the proposed consistent generalized spectral estimator to estimate a standard consumption-based CAPM, xing the potential lack of identi cation in Berkowitz (00). [Estimation results will be included.] 5 Concluding remarks This paper proposes a consistent generalized spectral estimator for models de ned by conditional moment restrictions. By employing the frequency domain approach, our estimator is close to a generalized spectral estimator proposed by Berkowitz (00), but resolves the lack of identi cation problem, caused by the global identi cation assumption in the GMM literature. Although Domínguez and Lobato (004) point out this issue and provide a consistent estimator in the time domain framework, their estimator has a serious drawback: incompatible with high dimensional data. In contrast, our estimator can be applied to the applications with high dimension data. However, our estimator has limitations of using pairwise dependence measures whereas it provides more precise estimation rather than using the whole information set. This is left for future research. 4

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