Testing the Permanent Income Hypothesis Using Taiwan Data: The Application of the FM-GMM and FM-GIVE Methods

31  Download (0)

全文

(1)ACADEMIA ECONOMIC PAPERS 27 : 1 (March 1999),49-79. it,. TESTING THE PERMANENT INCOME HYPOTHESIS USING. TAIWAN DATA: THE APPLICATION OF THE FM-GMM. AND FM-GIVE METHODS. Hsiu-hua Rau*. ABSTRACT. This paper deals with tests of the permanent income (PI) theory taking into account both the nonstationarity of income and consumption and the existence of adjustment costs ;t::§. of consumption. We carefully distinguish between the concept of actual consumption and. rp~. desired consumption. It is assumed that the desired consumption is a fixed proportion. 1t- 'it ;jJ: If ~J\fj. of permanent income, and that the actual consumption follows a quadratic optimization. rule. More specifically, we assume that consumers minimize the discounted sum of squared deviations of actual consumption to desired consumption and the square adjustment costs of actual consumption over time. In the general specification of the consumption function, consumption responds to the disequilibrium error due to the existence of adjustment costs as well as to changes in permanent income signaled by unexpected changes in income. Under the PIH, the sum of the coefficients of these two components is equal to one when the permanent income elasticity of consumption is unity. Tests of the PIH include tests of unit income elasticity and the linear constraint. The restrictions implied by the PIH correspond to the restrictions on the coefficients of 1(1) regressors that are possibly cointegrated. Furthermore, due to the fact that the error term in the consumption function is a moving average process, Johansen's procedure is not • Associate Professor, National Cheng-Chi University, Taipei, Taiwan. I would like to thank P.c.B. Phillips and two anonymous referees for comments on the paper..

(2) ACADEMIA ECONOMIC PAPERS 27: 1 (1999). 50. applicable. The asymptotic theory developed by Phillips and Kitamura (1993) allows us to conduct PIH testing using an inferential procedure based on FM-GIVE and FM-GMM. In this paper we found that the PIH is not rejected in Taiwan quarterly data. Under .. the maintained hypothesis of PI theory, the consumption function could be modeled as an ECM. Finally, even if we take into consideration the nonstationarity of consumption and income, if we neglect the problem of adjustment costs of consumption, the PIH would be rejected.. s I. Keywords: Permanent income, Consumption function, Nonstationarity, ECM, FM-GIVE, FM-GMM. t. c. I. 1.. INTRODUCTION. g e. d. The permanent income hypothesis (PIH) postulates that desired consumption is. a. a fixed proportion of permanent income. The difficulties in testing this hypothesis stem from the fact that both permanent income and desired consumption are. c,. rc. theoretical concepts and not directly observable. Ya-hwei Yang (1980) evaluated five methods for the estimation of permanent income, and tested the PIH. Tsong­ min Wu (1989) tested the PI theory taking into account the seasonal patterns and. p. the time trend of the consumption data. Neither author explicitly takes account of the nonstationary property of income and consumption in the estimation. In the consumption literature, Mankiw and Shapiro (1985) and Nelson (1985) pointed. IT. out that if income and consumption are nonstationary rather than stationary, then the standard testing procedures will tend to be biased toward the rejection of the PIH. However, West (1988), Campbell (1987), Campbell and Deaton (1989). T. and Campbell and Mankiw (1990) and Deaton (1992) showed that, even if the. it. nonstationary property of time series is taken into account, the PI model usually does not characterize the actual (U.S.) data, and Corbae, Ouliaris and Phillips (COP). tJ. 11. (1991) found that the PIH is partially rejected. Although these authors consider the issue of nonstationarity, they do not address the problem of adju'stment costs of. tJ. consumption. One important question is whether or not the failture to take account of the nonstationary property of income and consumption and adjustment costs of. T. consumption leads to an incorrect inference on the PI theory. Using Taiwan quarterly data, we show that if a testing procedure does not. o.

(3) TESTING THE fERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. 51. ;; to. simultaneously take into account the findings that income and consumption have unit roots and that consumption is slow to react to changes in permanent income due. Ider. to adjustment costs, then it may induce an incorrect rejection of the PIH. Under the maintained hypothesis that income and consumption have unit roots, we carefully. ; an and I be. distinguish between desired consumption (en and actual consumption (et). It is important to note that what the PI model predicts is desired cobsumption (or desired. 'VE.. saving) behavior which is not observable at least at the aggregate level. To test the PIH by comparing the actual data with what the PI theory predicts would require the assumption that the actual data coincide with the desired behavior. This is true only when an economy is in a steady state or when there are no adjustment costs. In general, there exist substantial transaction costs of adjusting stocks of durable goods. Moreover, as Bernanke (1985) showed, when durables and nondurables enter the utility function nonseparably, the effect of adjustment costs spill over from. n is. durables to nondurables. This implies that actual consumption usually takes time to adjust to the desired level and therefore actual consumption is not equal to desired. esis are. consumption. The difference between actual consumption and desired consumption reflects consumers' systematic adjustment behavior and cannot simply be attributed. lted mg­. to measurement errors. In order to distinguish desired consumption from actual consumption, in this paper we explicitly model consumers' adjustment behavior. It is assumed that desired. and ,unt In. .ted. flen. of. '89). the ally. )P). der ; of. unt : of. not. consumption is a fixed proportion of permanent income and that actual consumption follows a quadratic optimization rule. More specifically, we assume that consumers minimize the discounted sum of squared deviations of actual consumption to desired consumption and the squared adjustment costs of actual consumption over time. It is found that, due to adjustment costs, actual consumption is not a martingale. This implies that changes in actual consumption may be correlated with past income innovations as well as past income changes. We show that the PI theory implies that the coefficients of the consumption function should be subject to a linear constraint when there exist adjustment costS.1 As shown in COP (1991), the PI theory also implies that the MPC out of (zero frequency) income is unity. Our tests of the PIH thus include tests of the unit income elasticity and the linear constraint. Applying Taiwan quarterly data, we found that the PIH is not rejected. Moreover, it is pointed out that, under the maintained hypothesis of the PI theory, the consumption function Under the assumption that the marginal propensity to consume (MPC) out of (permanent) income is unity,.

(4) ACADEMIA ECONOMIC PAPERS 27 : 1. 52. of Taiwan can be modeled as an error correction model. This paper is organized as follows. Section 2 describes our models for testing the PI theory. Section 3 discusses our estimation methodology and empirical results. The FM-OLS procedure of Phillips and Hansen (1989) and the FM-GMM and FM-GIVE procedures of Kitamura and Phillips (1993) are suggested for conducting the test of the PIH. It is well known that in nonstationary regression models, OLS estimation suffers from second order bias problem. On the other hand, the fully modified OLS procedure (FM-OLS) of Phillips and Hansen (1989) solves the second order bias problem. However, FM-OLS does not allow for co integration in the regressors. In cases where the regression model contains both 1(0) and 1(1) regressors 2 or in cases where the regression model includes only 1(1) regressors that are cointegrated, we would apply FM-GMM and FM-GIVE for the estimation. (In the Appendix we provide the definitions and the empirical application of the FM-OLS, FM-GIVE, and FM-GMM estimators.) The implementation of the above FM estimations are conducted using Gauss procedure COINT 2.0 developed by Sam Ouliaris and P.C.B. Phillips. Section 4 concludes the paper.. 2.. MODELS FOR TESTING THE PERMANENT INCOME HYPOTHESIS. In this paper, we assume that the desired consumption c; is a fixed proportion of permanent income, i.e., c;. = (jyf + et,. where yf is permanent income, et is the. ¢ I t - 1 , i.e., et is in the information set in period t, but not in the information set in period t - 1. It is assumed here transitory component of c;, and et E It but. et. that Et-1et = 0 and E(.lt - 1) is defined as the conditional expectations operator. As in Flavin (1981), we define permanent income yf as the sum of capital income and the infinite horizon annuity value of human wealth: YtP -- Ytk+Ht. Here. yf. (1). is the capital income, and H t is the annuity value of human wealth. Real. human wealth H t is defined as the present discounted value of expected future labor 2. If some of the regressors are 1(0), then the regressors as a whole are trivially cointegrated, since any vector which puts non-zero weights on the 1(0) components and zero weights on the 1(1) components is a cointegrating vector..

(5) 53. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. income:. Ht = where. y;. r. +r. I(l+r)-kEtY~+k. (2). k=O. is the labor income, and r is the expected real interest rate which is. assumed to be constant. Denoted by. Wt.. the wealth at the beginning of period t,. then the capital income yf is the annuity value of wealth, i.e., yf. = rwt,. and. Wt. evolves according to. Wt. where. = (1 + r)Wt-l + yLl €r. Ct-l. + €r. is unexpected capital gains, and E t -. (3). 1. €'t'. O.. This paper is different. from much of the literature in that we explicitly distinguish desired consumption c; from actual consumption now explain.. Ct.. In general, c; is not equal to. Ct. Usually, the economy is not in a steady state.. for reasons we. This means that. consumers are revising estimates of permanent income from time to time, and hence c; is changing accordingly. If there are no adjustment costs and consumers are not subject to liquidity constraints, actual consumption will adjust to changes in the desired consumption instantaneously. In this case, the difference between. Ct. and c; is no more than a measurement error and the distinction is not significant. However, because of the existence of adjustment costs of changing durable stocks and the spill-over effect of adjustment costs from durables to nondurables, actual consumption does not fully adjust to changes in desired consumption. We cannot simply attribute the difference to measurement errors or any serially uncorrelated error process.. It also reflects consumers' adjustment behavior.. It is therefore. important, when applying the actual aggregate data to test the PIH, to distinguish actual consumption from desired consumption. To explicitly model consumers' adjustment behavior, we assume that actual consumption follows a quadratic optimization rule. More specifically, we assume that a representative consumer solves the following problem to achieve the desired consumption level over time 00. min E t Ct. 2:. s=t. o;s-t[(1(Ct. cn. 2. + (Ct. ct_d 2 ]. (4).

(6) 54 . ACADEMIA ECONOMIC PAPERS 27 : 1 (1999). where. 0:. is the discount factor (0 <. 0: :::;. 1) which does not have to be equal to. T,. and the expectation is taken with respect to the information set at time t. (4) is a quadratic loss function which penalizes both deviations of consumption. Ct. from. the desired level c; (the first term of (4» and movements in consumption itself (the second term of (4». penalty of deviation of are small (large). nondurables. 3. Ct. In (4), (3 represents the relative weight put on the from. c;.. If (3 is large (small), then adjustment costs. In general, durable goods have higher adjustment costs than. There is a simplification involved in using (4). Our justification is. as fonows. Since large scale changes in consumption involve a higher proportion of changes in the consumption of durables, they are usually accompanied by large amounts of adjustment costs. Therefore the quadratic loss function takes account of the adjustment costs difference between durables and nondurable goods.. Remark: It is a standard practice in the consumption literature to use equations (1) to (3) or their variants to derive implications of the permanent income (PI) theory and test them with the actual data. For example, by applying (1) to (3), Flavin (1981) analyzed the role of current income in providing new information about future income under the PIH. Campbell (1987), using (1) to (3), derived "savings for a rainy day" implications of the PI theory. We note that what the PI model (equations (1) to (3)) predicts is the behavior of desired consumption or desired saving which is unobservable. To replace the unobservable variables with the actual data to conduct hypothesis testing requires the assumption that the actual data coincide with the desired behavior (i.e., the actual data always satisfy a unit PI elasticity restriction). As argued above, this is valid only when the economy is in a steady state or when there exist no adjustment costs. The optimization problem of (4) has a forward solution given by. Ct =. ,/Ct-l. + (1 -. 00. ,/)(1 - o:(3)Et. L (o:,/)S-t c;. (5). s=t. where '/ is the stable root of the quadratic equation o:z2 - (1 3 Justifications. + 0: + (3)z + 1. O.. for the existence of adjustment costs of nondurable goods are given in Bernanke (1985). With the assumption that nondurables and durables enter the utility function non-separately, and that there exist adjustment costs for durables, Bernanke (1985) showed that the effect of adjustment costs can spill over from durables to nondurables..

(7) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. r,. Since yr is 1(1) and Et-I(~Yn. 55. 0,4 we derive the partial adjustment model. is m. Ct = ,Ct-I. + (1 - ,)Oyr + (1 - cl:f)(I- ,jet. (6). If Ie. Equation (6) can be rewritten as. ts. m is. ~Ct = (r - I)(Ct-1 -. Oyr-I). + (1 -. ,)O~yf. + (1 - cq)(1 - ,jet. (7). m ge. which offers the convenient interpretation that agents marginally adjust actual con­. nt. sumption Ct from ~yr. 3). nd. :1). Ire. 'a. lOS. I. is. lct :he n).. len. Ct-l. in response to changes in (current) permanent income. «(1 - ,)0 being the short run effect), and the previous "disequilibrium error". (OyP C)t-l «(r 1) being the feedback coefficient V The presence of the "dise­ quilibrium error" is due to the lag adjustment of the actual consumption to the desired level. 6 Equation (7) is of error correction form and is similar to the annual version of the consumption model in Davidson et aL (1978) (with the inflation terms omitted) except that the income variable that appears in (7) is permanent income instead of disposable income. Although the difference seems minor, it implies a fundamental difference regarding the assumption of consumers' behavior. Whereas our approach involves forward-looking consumption behavior, i.e., our agents adjust their consumption to estimates of future labor income, Davidson et al. (1978) interpret their error correction model in terms of the disequilibrium adjustment of consumption to income. This also implies that our model satisfies a unit permanent income elasticity restriction instead of a unit disposable income restriction. Alternatively, (6) can be reparameterized as Ct = aO~Ct. + Oyr + (1 - a,)et. (8). (5) where ao. : O.. '85). and aent. 4. 5. 6. 1-,. The advantages of (8) over (7) is that the equilibrium. With the assumption that E t _ 1€t = 0, E t _ 1(Llyn = 0 because yi is a martingale as shown in Flavin (1981). It is not surprising that for intertemporal optimization problems such as (4), the error correction model of (7) results. As shown in Nickel (1985), Euler equations derived from quadratic objective functions could be recast as error correction models. When there are no adjustment costs, Ct in (7) is equal to c;, and (7) yields an equilibrium solution, c; 9yf + et. This is true since in this case, f3 -t 00, so 'Y -t 0 in (7). i.e., ct.

(8) 56 . ACADEMIA ECONOMIC PAPERS 27: 1. coefficient 0 and the short run coefficient Qo are linear and therefore can be estimated simultaneously.. However, direct estimation of (8) is difficult since. yf. is unobservable. The estimation strategy we use in this paper avoids the explicit treatment of yf. The basic idea is to exploit the following stochastic difference equations for human wealth and capital income 7. (9) (10) r. where E~ is equal to. ~. £oJ. + r k=O. (1. + r) -k (Et. -. I Et-dYHk,. representing the unfore­. castable revision from t - 1 to t in the expected value of human wealth, tf' is the. = 0 = Et-1t¥'.. unexpected capital gains, and Et-lt~ the definition of. yf. yf. (9) and (10) corne directly from. (3) and of H t (2). Now using (9) and (10), we can eliminate. from (8) to obtain. Ct. = (1. r(1 - 0) . 0. w. h. Vt. + --1--- )Ct-l + ,(1 + r)~Ct-l + -1--(rf t + Et ) + -1·--~. where Vt = (1 - Q,)[et - (1. (11). -~-~. + r)et-l]'. Hall (1978) showed that consumption (or, more strictly speaking, desired con­. t. sumption) must follow a martingale if the life cycle-permanent income hypothesis is true. Since then, tests of the PIH have been frequently formulated as tests of whether or not c; is a martingale, or equivalently, as tests of whether information available at time t - 1 other than. c;_l. is useful to predict. c;.. The difficulty of. testing whether c; is a martingale sterns from the fact that the data of c; is normally not available at the aggregate level. A common practice is to use for c; and then test whether rejected if. Ct. Ct. as a proxy. is a martingale: the permanent income theory is. is not a martingale. (See, for example, Flavin (1981), Bernanke(1985),. Blinder and Deaton (1985) among others).8 In fact, 7 Similar. Ct. Ct. is not a martingale even if. ideas have been developed in Hayashi (1982). more sophisticated proxy used in the literature is actual expenditures on nondurables. The validity of this procedure requires two conditions. First, desired nondurable consumption is a fixed proportion of total desired consumption. Secondly, the desired consumption on nondurables is equal to the actual consumption on nondurables. The first condition is not satisfied if nondurables and durables enter the utility function nonseparably. The second condition holds only if there exists no spillover. 8 A. t.

(9) 57. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. ~. e=. be. the PIH is true. To see this, since the permanent income theory implies that. yi. as demonstrated in Corbae, Ouliaris, and Phillips (COP) (1991),9 (11) becomes. ,licit. (substituting -,_1_ for 1. 1. ao in (11». ~(. :nce ~Ct = ,(I. (9). + r)~Ct-l + (1. ,)(rE;". + E~) + (1. (12). ,)Vt. or equivalently. 10) ~Ct = (1. -,(1. + r)L)-l(1 ,)(rc;" + E~) + (1 -,(I + r)L)-l(1_,)Vt. (13). )re­ Equation (13) indicates that changes in consumption is a weighted average of the. past income shocks (which induce consumers to revise their estimates of permanent. 'om. income). This suggests that. late. a martingale under the PIH due to the existence of adjustment costS.lO Supposing. ~Ct. is not a martingale difference and hence Ct is not. that the PIH holds and there are no adjustment costs (r ~Ct. :11). rE;". + E~ + Vt. = 0),. (12) is reduced to. (14). which is the consumption equation derived by Flavin (1981). If we assume further on­. that the transitory consumption is equal to zero (i.e., et = 0), then (12) becomes. esis , of. A. UCt. ion of ally oxy t. =. UJ. rEt. + Eht. (15). which is the Hall's martingale equation for consumption. Solving (10) for rE;" and then substituting it into (15), we obtain. is. 15), 1 if. dity tion the tiles Iver. St -. 9. 10. ~y~ - (I. + r)St-l. = -E~. (16). effect of adjustment costs from durables to nondurables. Another approach is to convert data on expenditures on durables goods into a series of imputed service flows. However, this approach neglects the existence of adjustment costs. They demonstrated that the marginal propensity to consume out of zero frequency income is unity. (12) also implies that b.Ct follows ARMA(l,l). Such an ARMA{l,l) consumption function could arise from the optimal consumption rule as Mankiw (198]) suggested..

(10) ACADEMIA ECONOMIC PAPERS 27 : 1. 58. where. St. = y~ - Ct. Equation (16) is exactly the dynamic restriction imposed by the. PIH that is derived by Campbell (1987). Equation (12) is, in effect, a linear combination of the partial adjustment model and the martingale equation.. The first term on the right hand side of (12) is. + r}Act_d, and the (ref + f~ = Ayf). The. related to the compound value of the previous adjustment «(1 second term is related to changes in permanent income. effects of the previous adjustment and changes in ,(current) permanent income on changes in current consumption are represented by "Y and 1 - "Y, respectively. When "Y = 0 (adjustment costs do not exist), actual consumption Ct fully adjusts to desired. consumption. c;, the first term on the right hand side of (12) vanishes, and changes. in (current) consumption are fully explained by changes in (current) permanent income. When "Y =I 0 (there exist adjustment costs),. Ct. does not fully adjust to. c; and the first term reflects "left-over" effects due to the existence of adjustment costs. The larger "Y is, the more significant the first term becomes, which means that more of ACt is accounted for by the left-over effects. More importantly, the sum of the coefficients of the first term and the second term is equal to one independent of the magnitude of the adjustment costs.. Thus, we derive a test of the PIH. when adjustment costs exist. In the consumption literature, tests of the martingale hypothesis are equivalent to tests of whether the coefficient of the first term is equal to zero and the coefficient of the second term is equal to one, which does not take account of the existence of adjustment costs. Such a test would be more likely to reject the PIH when adjustment costs are high ("Y tends to be away from zero) and even if the PIH is true. lt is important to point out that to test whether 0 = 1 in (11) is not equivalent to. the test of the PIH. When 0. 1, (11) implies that Ct is 1(1). However, an arbitrary. 1(1) process for consumption does not in general satisfy the constraint that the expected present value of consumption is equal to the total human and nonhuman wealth. The permanent income theory also suggests that, when adjustment costs exist, the regression coefficients in (12) are subject to a linear constraint as we discussed above. We note that the validity of 0 = 1 should be tested before assuming (12) and attempting to test therein Ho: the sum of the regression coefficients in (12) is equal to one. Equation (13) implies that consumption is not only related to current income changes but also past income changes. Nevertheless, the relationship is subject to constraints when the PIH is true. In particular, the PIH suggests that only unan­.

(11) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. the. 59. ticipated components of current and past income changes can affect consumption. Because income changes include anticipated and surprise components, statistically,. del. consumption may still be correlated with past income changes even if the PIH holds.. is. The correlation between consumption and past income changes can be explained as. I. the. follows. Suppose, in period t. fhe. to an unexpected increase in wealth). This raises desired consumption. Because of. 1, the consumer experiences a change in PI (say, due. on. adjustment costs like transaction costs, search costs, or the lack of perfect capital. hen. and future markets, actual consumption during the period of the shock will not. red. bring the consumption to the desired level. Instead, a gradual increase in stocks. 1ges. over the future will be anticipated. Therefore, it is reasonable to say that part of. lent. the actual consumption expenditure in the future period (e.g., in period t) accounts. . to. for the changes in the desired consumption in the previous periods, which are, in. lent. turn, statistically related to past changes in income. Because actual consumption is. that. slow to adjust to changes in income innovations, changes in consumption are thus. of. related to past income changes. The fact that actual consumption is excessively. lent. sensitive to income may well be because of adjustment costs rather than liquidity. PIH. constraints. As a consequence, statistically, it is difficult to distinguish the effect. gale. of liquidity constraints from that of adjustment costs on consumption. A valid test. 1. lual. of the PIH therefore is not simply testing whether current actual consumption is. :ake. related to changes in past income. One advantage of our framework is that we can. y to. incorporate the equilibrium coefficient () into our regression model and conduct a. and. straightforward test of the PIH by concentrating on the coefficient (c.f. COP(1991)).. It to rary the. man xist,. ssed (12). l). is. ome. ;t to. Remarks: 1. Assuming that nondurables and durables enter the utility function nonsepara­ bly, and that there are adjustment costs of durables, Bernanke (1985) also showed that consumption does not follow a martingale. Although Bernanke's model implies a partial adjustment mechanism similar to ours (with the assumption that et the. ci. 0),. implicitly implied in his model is a fixed proportion of human wealth, not. permanent income. 2. One of the fundamental differences between our regression model (11) and most of the Euler-type specifications generated from household intertemporal optimization problems in the consumption literature (with the exception of COP (1991»11 is that the latter usually do not define desired consumption as a fixed. nan11. However, the COP (1991) model does not take account of adjustment costs..

(12) ACADEMIA ECONOMIC PAPERS 27: I. 60 . proportion of permanent income. Consequently, it is difficult to say whether or not we should reject the PI theory if the data are not consistent with these Euler-type. of. models.. Wt. wi. M. 3. ISSUES OF ESTIMATION AND EMPIRICAL RESULTS. ill' In this section, we intend to use actual data to test the PIR. As mentioned in the last section, the permanent income theory is not rejected if () = 1 and the sum of the regression coefficients on (1 + r )LlCt-l and. (rEr. + E~). in (12) is equal to one.. ha. Pt. wi,. Since the validity of (12) is based on the assumption that () = 1, we conduct our tests of the PIR by first testing whether or not 8 not the coefficient of. Ct-l. 1, or equivalently, whether or. in (12) is significantly different from one. The regression. COl. an. model of (11) is. Ph (17). hY1. coi h were. ao. . h IS t e constant term, al. = 1 + r(1 1. 8). = a2 UCt-1 J\. C. - 0;0 . ,E t. + -1-- (w rEt (). 0;0. + Eh) + t. + r) and Vt is defined as before. Basically, (17) is a co integrated. -Vt 1 , az = ,(I 0;0. autoregressive model with a unit root. To see this, we note that (17) is observationally equivalent to. Ct. ao +. 1 [. -0;0(1. + -_... in which the regressors. + r) + r(1 1. Ct-l. 8) 1. Ct-l. 0;0 . and. Ct-2. +. 0;0(1. 1. + r). - 0;0. imf Ct-Z. are cointegrated.. +. Vt. (18). - 0;0. be FM. (18) is related to (17). through the "co-ordinate rotation" method of Park and Phillips (1988, 1989). In general, under the assumption that the cointegrating vector. 0;. (1. Ct. is 1(1),. 1), so that. consumption innovation. The relation {31 Ct-l (32)Ct-l. Ct-2),. (32(Ct-l. respectively,. 1(1). and. Ct-l Ct-l. + {32Ct-2. Ct-2. = Ct-Z. (31. + {'~,. where. t is the. E. can be re-expressed as ({31. where the two "new" regressors. and 1(0). In the present context,. are co integrated with. Ct-l. and. = 1 + -0;0(1. LlCt-1. ~ r) + r(1. +. are,. (In hyp'. 8). 0;0. and {32. 0;0(1. 1. + r). -0;0. ' so that {31. + f3z. = al, f3z. -az, and (18) is equivalent to (17).. 12 (.

(13) 61. TESTI!'lG THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. To test Ho : of. al. e=. 1 (or Ho :. al. = 1),. it would be useful to obtain an estimate. that has a standard limiting distribution permitting classical inferences. It is. well known, however, that, although the OLS estimate of the coefficient associated with the 1(1) regressor Moreover, because. Vt. Ct-l. is consistent, its asymptotic distribution is nonstandard.. is MA(l) and hence. E(Ct-lVt). =I- 0, the limiting distribution. involves nuisance parameters which make inference problematical. On the other hand, the modified test statistics on the OLS estimate of. al. (al) proposed by. Phillips and Perron (1988) have asymptotic distributions that are invariant within a wide class of weakly dependent and possibly heterogeneously distributed innovations. {Wt}r. The consumption and income data used in this paper are taken from the National Income Accounts of the Statistics Bureau. Consumption is the household consumption and income is GDP. Both variables are in constant dollars. All data are quarterly series (seasonally unadjusted) for the period 1961:1. 1996:4.. A. Phillips and Perron unit root test indicates that there is no evidence against the null hypothesis that. al =. e = 1 is thus not rejected.. 1. The assertion that. If we suppose it is known that e = 1, hence Ct is 1(1), and Ct-l and cointegrated. As mentioned earlier, in this case, (18) can be rewritten as. Ct. where •. = ao. PI. + (131 + ,82)Ct-l -. = 1+. -ao(l. Ct-2. Vt. + 1 -ao. P2 AC t-l. + 1') + 1'(1 e) . ~.-1 - ao. and 132. are. (19). ao(l + 1') = ----.-. 1 - ao. As expected,. e= 1. implies that 131 + 132 = 1. That is, the estimate of the coefficient of Ct-l should not be significantly different from one if the null hypothesis that e = 1 is true. The FM-OLS estimation of (19) is shown as follows:. Ct. = 11996 + (0.93). 0.958 Ct-l (15.45). -. 0.839 (3.61). (20). ACt-l. (In parentheses, we report the t-values of the respective estimates.) hypothesis that the coefficient of. Ct-l. is equal to one is not rejected.. The null. 12. Alternatively, since equation (17) is observationally equivalent to (18), we may 12. One of the drawbacks of the FM-OIS method is that the FM-OLS method treats stationary variables as potentially 1(1) variables in the estimation. This may not cause much problem if the sample size is large..

(14) ACADEMIA ECONOMIC PAPERS 27 : 1. 62. test Ho :. e=. 1 by estimating equation (18) to see if the sum of the regression. coefficients is equal to one. The regression model of (18) involves Ct-l and Ct-2 as explanatory variables that are cointegrated. The reason why the estimation of (18) is a problem here is because the OLS and IV estimators of the 1(1) coefficients of (18) have nonstandard distributions which makes statistical inference problematicaL. Various estimators have been proposed for which inferences can. be based on standard asymptotic distributions. These estimators include the fully modified OLS (FM-OLS) estimator of Phillips and Hansen (1990), the FM-GMM and FM-GIVE estimators of Kitamura and Phillips (1993) and the restricted system vector autoregressive (VAR) estimator of Johansen (1988).. Different from the. FM-IV methods, Johansen's procedure requires the assumption that there exists a finite order VAR representation for changes in consumption.. As seen from. (11), owing to the MA(l) in the error term, there is no finite order VAR for changes in consumptionY In this case, applying Johansen's method would involve misspecification for the data generating processes.. On the other hand, the FM. procedure is nonparametric, and therefore avoids the misspecification problem. Kitamura and Phillips (1993) develop the as),mptotics of FM-IV procedures for the case where the regression model involves cointegrated regressors such as (18). The FM-IV methods have an advantage over the full-system ML procedure because the application of the latter requires explicit construction of the likelihood function that is complicated in the case of MA(l) errors. In this section, we will adopt the FM-GMM and FM-GIVE procedures of Kita­ mura and Phillips (1993) for the estimation of (18). The empirical implementation of the FM-GMM and FM-GIVE methods are described in the Appendix. The FM estimations are conducted using the Gauss procedure COINT 2.0 developed by S. Ouliaris and P.C.B. Phillips. Table 1 and Table 2 report empitica\ tesu\ts 1.0\' the FM­ GMM and FM-GIVE estimations of (18), respectively. The test of Ho : fJl. + fJz = 1. is based on the Wald criterion. The rows give empirical results for various choices of instruments. (In parentheses we report the t-values for the respective estimates.) These instruments include 1(1) processes generated from random numbers (rndn), the lagged money supply (M1A), the lagged government expenditures (CG) and the lagged consumption (C). CG is taken from the National Income Account of the Statistics Bureau, and M1A is taken from the Aremos Data Bank.. 13 . This also implies that the lagged consumption cannot be used as instruments in the estimation of (18)..

(15) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. ion. 63. Table 1. Regression Results (FM-GMM). t-2. ion. Rows. IV. Ct-l. Ct-2. Test of restriction. (1). 1. rndn. 1.96. -0.965. not to reject. (1.58). (0.742). 0.62. 0.33. Ice ;an. 2. .M1A t -. ll •••. 1. M1A t _ 4. I. Illy. [M. (19.11) (10.27) 3. ;,m. the. ists. not to reject. 4. CG t -51' .. 1 CG t - 8. I. Ct -61 ... , C t -9. ;)m. 0.69. 0.29. (2.73). (1.12). 0.664. 0.263. (23.05). (9.06). not to reject reject. for. Table 2. Regression Results (FM-GIVE). lve ~M. :m.. Rows. IV. Ct-l. Ct-2. Test of restriction. he. 1. rndn. 5.977. -5.01. not to reject. (6.86). (5.70). 2.21. -1.2. (6.68). (3.52). 1.13. -0.136. (5.28). (0.62). 1.756. -0.944. 'he. he. 2. M1A t _ 1 , " ' , M1A t _ 4. !at 3. CG t ­. 5 , ••• ,. CG t ­ 8. ta­. :m M. 4. Ct -6, ... , Ct -. 9. 1. not to reject not to reject reject. (58.35) (31.46). S.. .1­. 1. The FM-GMM estimation of (18) shows contradicting results depending on the. es. instruments employed. The reason for this is that not all instruments used here. ;. ). are valid instruments. The validity of these instruments is thus tested using the. I),. FM-GMM and FM-GIVE validity tests.. Ie. statistics used to conduct the FM-GMM and FM-GIVE validity tests.) Among the. Ie. instruments used, only random numbers are valid instruments, and all the others are. )f. (In the Appendix we explain the test. not. If valid instruments are employed, the FM-GMM estimation shows that the null hypothesis, i.e., that the sum of the coeffcients is equal to one or that. e = 1, is. not rejected. As for the estimation of (18) using the FM-GIVE procedure, we first.

(16) ACADEMIA ECONOMIC PAPERS 27. : 1 (1999) .. 64. note that, except for the lagged consumption, all the other instruments used are valid. Table 2 shows that the null hypothesis that Ho : /31 + /32. = 1 is. not rejected. when valid instruments are used in the FM-GIVE estimation of (IS). On the other hand, when the instruments adopted are not valid, the FM-GIVE estimation tends to reject the null hypothesis, which is not, of course, a reliable conclusion. To sum up, to test the null hypothesis that (}. = 1, different estimation procedures. are used depending on whether or not the information that. and. Ct-2. are. cointegrated is available. If the information on the co integration between. Ct-l. and. Ct-2. Ct-l. is unknown, then the FM-OLS estimation of (17) and the FM-GMM and. FM-GIVE estimation of (IS) indicate that the null hypothesis that () rejected. On the other hand, if (}. 1 is true, hence. Ct-l. and. Ct-2. =1. is not. are co integrated,. the regression result from the estimation of (19) by FM-OLS further confirms the fact that (} = 1. When (). 1, the test of the PI theory requires the estimation of (12) and the. test of whether or not the sum of the regression coefficients in (12) is equal to one. Equation (12) is reproduced here as. (21). where bo is the constant term, b1 I, b2 = 1- I and b1 + b2 = 1. A test of the linear constraint on individual coefficients in (21) requires two things. First, it requires that one specify or estimate an interest rate r. per cent on an annual basis.. 14. For simplicity, we set r. 4.04. However, our testing results are not sensitive to. the choice of r. Second, it requires that we develop a proxy for the explanatory variable Tlo,/,. + c~. because it is not observable. In effect, TC,/,. + c~. represents an. unanticipated component of changes in disposable income. In this paper we use changes in GDP as a proxy for the changes in disposable income. Since an AR(S) process is sufficient to describe changes in GDP, we will use the residual from the regression of D.Yt on D.Yt-l, D.Yt-2 ... D.Yt-8 as a proxy for rc,/,. + c~.. Equation. (21) is estimated by the two-stage least squares (TSLS) and GMM methods.1 5 The instruments adopted include the difference in government expenditure, changes in money supply MIA and saving (Y - C). The instruments chosen here are among 14. 15. Campbell (1987) also set r = 4.04 per cent. Dummy variables are included in the regression equation to take account of seasonal factors. However, the estimates of the coefficients of the dummy variables are not reported in Table 3..

(17) 65. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. re. those that are commonly used in the consumption literature. Since our result is. ~d. not affected by the choiee of r, we only report in Table 3 the regression result for. er. (21) under the assumption that r. ds. the regression result when we estimate the equation (21) by OLS. The remaining. 4.04 per cent. The first row of Table 3 shows. rows give TSLS and GMM results for the case when the instruments chosen are the es. lagged saving and one period lagged difference in government expenditures. The. re. last column gives the results for testing the null hypothesis that b1 + b2. Id. test is based on the Wald criterion. (In parentheses we report the t values for the. ld. respective estimates.). 1. This. )t. Table 3. Regression Results (r. d,. 4.04 per cent). Ie. Rows. Methods. Constant term. (1+r)~Ct_l. Ur+f~. Test of restriction. 1. OLS. 35125.37. -0.85. 0.25. strongly reject. (12.36). (20.74)**. (1.46). 38319.5. -0.86. 1.77. (6.09). (10.66)**. (1.84)*. 28366.16. -0.78. 1.17. (7.34). (14.38)**. (1.38). Ie. e. 2 3. l). lr. TSLS GMM. * means that the coefficient is significant at. 0:'. ** means that the coefficient is significant at. 0:'. not to reject not to reject. = 10%.. 5%.. :s. The TSLS estimates of the coefficients of (1. 4. + r)~Ct-l. and. ur + tf. are both. ). significantly different from zero at the 10% critical level.. Moreover, the TSLS. y. regression result shows that the null hypothesis Ho : b1 +. 1. This implies that, under the assumption of the PIH, though consumers adjust their. ~. consumption in response to unexpected changes in income, the adjustment is slow. ). due to the existence of adjustment costs. Similarly, the GMM estimation of (21). ~. 1 is not rejected.. does not reject Ho : b1 + b2 = 1 (or the PIH). However, the estimate of the coefficient of. ur + tf. is not significantly different from zero. The conclusion that the PIH. is not rejected remains valid if we choose a different set of instruments or include more variables in (21). For example, suppose that we add an explanatory variable ~Yt-l. to (21), and estimate the resulting regression model by both the TSLS and. GMM methods. The instruments used are the same as those used for the estimation of (21). The empirical results are summarized in Table 4..

(18) ACADEMIA ECONOMIC PAPERS 27 : 1. 66. Table 4. Regression Results i. Rows Methods Constant term (l+r).6.c t _l rE~+E~ 1 2 3. OLS. Test of restriction strongly reject. 22802.73. -0.88. 0.19. 0.64. (9.76). (29.54)**. (1.55). (10.86)**. 47.98. -0.94. 2.16. 1.008. (0.003). (10.75)**. -1539.334. -0.93. (0.12). (11.76)**. TSLS GMM. .6.Yt-l. (2.13)** (3.55)** 2.39. * means that the coefficient is significant at. Q. ** means that the coefficient is significant at. Q. not to reject. 0.99. not to reject. (3.46)** (4.12)** 10%. = 5%.. As shown in Table 4, the null hypothesis that the sum of the coefficients of (1 + r).6.Ct_l and rE~ + E~ is equal to one is tested based on the Wald criterion. The result shows that the restriction is, again, not rejected. Because the estimates of the coefficients of. .6.Ct-l. and .6.Yt-l are significantly different from zero, consumption. is not a martingale. We note that our basic results are not affected by the change in the list of instruments. As for the estimates of the coefficients of (1 + r).6.Ct_l and (rE~. + E~),. the TSLS estimation gives similar results as before. Different from. the previous case when the regression model (21) is estimated without including .6.Yt-l as an explanatory variable, the GMM estimate of the coefficient of (rE~ + E?). v v. becomes very significantly different from zero at the 5% critical ievel.t 6 From the above results we infer that, taking into account the existence of. d. adjustment costs of consumption and the nonstationarity property of consumption. a. and income, the PIH is not rejected in Taiwan quarterly data.. (. Furthermore,. consumption is not a martingale even if the PIH is true. To verify this further, we consider the following regression model:. p p. (22). Substracting. 16. Ct-l. from both sides of (22), we obtain. We found that the estimate of the coefficient of variables.. rtf + Ef. is sensitive to the choice of instrumental.

(19) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. 67. The PIH implies that, in (23), d2 + d3 = 1 and d2, d3 and d4 are significantly different from zero. In the consumption literature, if the coefficients of ~Ct-l or ction. Yt-l. lect. the PIH. As was discussed in the last section, d2. :ct. the PIH if there are adjustment costs. This only reflects the fact that consumers are slow to react to changes in permanent income (due to the existence of adjustment costs) and hence the nonzero coefficients cannot be considered as evidence against. :ct. are significantly different from zero, this would be regarded as evidence against. t. 0 or d4. t. 0 can be consistent with. the PIH. We note that when d 1 and when there are no adjustment costs, (22) becomes (setting d2 = 0 = I and d 1 1 in (22» ~Ct =. do. +. d3(rE'f. + E~) +. d4Yt-l. + Ut. (24). lts of. . The )f the. which is, in essence, the regression model Flavin and others used to carry out the excess sensitivity test (under the assumption that there are no adjustment costs).. Iption. Under the same condition, equation (22) can also be rewritten as. lange ~Ct-l. Ct =. (25). do + d1Ct-l + d4Yt-l + u;. from. u;. uding. where. +E~). with one-period lagged income in Hall (1978).. ce of. To verify the PIH, we need to show that d2 + d3 = 1 and d2 , d3, d4 are significantly different from zero. The regression model of (22) involves income and consumption. d2 (1 + r)~Ct-l + d3(TE'f + E~) + Ut. Equation (25) is the regression model. lption more,. (18), (22) also includes two 1(0) variables ~Ct-l and rE'f. :r, we. ways to conduct the estimation of (22). The first method is to apply the FM-OLS. as explanatory variables that are cointegrated. Different from the regression model. + Ef.. There are several. procedure directly for the estimation of (22), treating 1(0) variables as 1(1) variables. Alternatively, we can apply FM-OLS or FM-GlVE to (25), and obtain the residual (22). (23) mental. it;.. If the residual is 1(0), then we run the regression of. TE'f. + Ef.. it;. on (1 + r )~Ct-l and. Using the first method, the following regression result is obtained:. Ct=-2796.34+ 0.70 Ct-l- 0.79 (1 + r)~Ct-l+ 0.23 (TE'f + E~) + 0.18 Yt-l (1.49) (11.99) (29.91) (0.91) (5.52) The FM-OLS estimation rejects the null hypothesis that d2 + d3. (26). 1. However,.

(20) 68. ACADEMIA ECONOMIC PAPERS 27.' 1. from (25), the sign of the constant term is negative. This implies that the FM-OLS method may not be reliable when the regression model contains both 1(0) regressors and 1(1) regressors that are cointegratedP Using the second method, we first run the regression of. Ct. on. Ct-l. and. Yt-l'. (That is, we first estimate the model (25). (without the constant term) by FM-GIVE.) We obtain the following result:. Ct. 1.0022 Ct-l (19.9). + 0.00276 Yt-l. (27). (0.098). The FM-GIVE estimation of (25) shows that the coefficient of one-period lagged income is not significant. This is due to the fact that the constant term is excluded in the estimation. If the regression model includes a constant term, the estimate of the coefficient on one period lagged income is 0.088, and the corresponding t value. hyp<. is 2.15, which is significantly different from zero at the 5% critical level. lS Hence. one. the current consumption is related to past income. According to the Phillips-Perron. 1. unit root test, the residual from the above regression result is 1(0). Hence, treating the residual as a dependent variable and (1. + r)~Ct-l. and. rfr + E~l. and. as explanatory erro. variables, we run the following regression:. (28). + d3 = 1.. Table 5 reports the empirical results for the. (29). OIS, TSLS and GMM estimations of (28). (The instruments used are the same. vari~. as those used for the estimation of (21).) The second, third, and fourth columns. not. and test whether or not d2. give the estimates of the coeffcients corresponding to the constant term,. rfr + E~.. ~Ct-l. and. The last column gives the result for testing the null hypothesis that the. sum of the coefficients of (1. + r)~Ct-l. and rf~v. + E~. is equal to one. This test of. the restriction is based on the Wald criterion.. the estin und( Taiw. The regression results as shown in Table 5 are very similar to those in Table 3.. Taiw. From Table 5, the estimate of the coefficient of (1 + r)~Ct-l is significantly different. (29). from zero. This implies that consumption is not a martingale. Moreover, the null. of tl. 17. 18. In this case, the cointegrated regrssors are Ct_1 and Yt-I' This implies that, if we test the PIH ignoring the existence of adjustment costs, we would tend to reject the PI theory..

(21) 69. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. LS. Table 5. Regression Results. ors 'un. Rows. Methods. Constant term. 6.Ct-l. rEI"+E~. Test of restriction. 25). 1. OLS. 36113.90. -0.86. 0.26. strongly reject. (12.41). (20.22)-­. (1.47). 39262.9. -0.86. 1.87. (6.03). (10.29)**. (1.88)*. 28119.21. -0.75. 1.08. (7.05). (14.25)*­. (1.16). 2. TSLS. 27) 3 ged jed ~. GMM. * means that the coefficient is significant at ** means that the coefficient is significant at. not to reject not to reject. a = 10%. a = 5%.. of. + r)6.Ct_l. due. hypothesis that the sum of the coefficients of (1. nce. one is not rejected. Hence the PI theory is not rejected.. and rEI". + E~. is equal to. 'ron. The reason for the similarity between the regression results as shown in Table 5. ting. and the regression results as shown in Table 3 is explained as follows. Substracting. tory. Ct-l from both sides of equation (22) and rearranging terms, we obtain the following. error correction model (ECM):. (28). the. 6.Ct. do - (1 - dd[Ct-l - 1 d4d1 Yt-l]. + d2(1 + r)6.Ct_l + d3(rE~V + E~) +Ut (29). (29) is similar to (21), except that the former now includes one more explanatory 1 ~4dl Yt-d· Since, as mentioned earlier, the estimation of (21) is. ime mns. variable. and. the empirical results for the estimation of (21) are very similar to those for the. the. estimation of (21). Since (22) or (29) is consistent with the PIH, this implies that,. of. under the maintained hypothesis of the PI theory, the consumption function of. t. e 3. rent null. Idto. [ct-l -. not affected by adding more variables to equation (29), it is not surprising that. Taiwan can be modeled as an error correction model. The consumption function of Taiwan studied by Lee and Wang (1994) is also an error correction modeL However, (29) contains unexpected changes in income which is excluded in the specification. of the consumption function of Taiwan studied by Lee and Wang (1994)..

(22) ACADEMIA ECONOMIC PAPERS 27: 1. 70. 4.. CONCLUSION. In this paper we derive econometric models to test the permanent income theory taking account of the existence of adjustment costs of consumption and the nonstationary properties of income and consumption. We show that, under the permanent income hypothesis (PIH), actual consumption is not a martingale process when adjustment costs exist. In the general specification of the consumption function, consumption responds to the disequilibrium error (lag adjustments) due to the existence of adjustment costs as well as to changes in permanent income signaled by unexpected changes in income. Under the PIH, the sum of the coefficients of these two components is equal to one when the (permanent) income elasticity of consumption is unity. Tests of the PIH include tests of unit income elasticity and the linear constraint. The asymptotic theory developed in Phillips and Hansen (1989) and Kitamura and Phillips (1993) allows us to conduct PIH testing using inferential procedures based on FM-OLS, FM-GMM and FM-GIVE. It is found that the PIH is not rejected in Taiwan quarterly data. Moreover, under the PIH, the consumption function of Taiwan can be modeled as an error correction model.. REFERENCES. Bernanke, Ben (1985),. '~djustment. Costs, Durables, and Aggregate Consumption," Journal. of Monetary Economics, 15, 41-68. Campbell, J.Y. and A. Deaton (1989), "Why is Consumption so Smooth," Review of Economic Studies, 56, 357-374. Campbell, J. Y. and N. G. Mankiw (1990), "Permanent Income, Current Income, and Consumption," Journal of Business and Economic Statistics, 8, 265-279. Corbae, D., S. Ouliaris, and P. C. B. Phillips (1991),. '~. Reexamination of the Consumption. Function Using Frequency Domain Regressions," Cowles Foundation Discussion Paper No. 997. Davidson, J. E. H., D. F. Hendry, F. Srba and S. Yeo (1978), "Econometric Modeling of the Aggregate Time-Series Relationship between Consumers' Expenditure and Income in the United Kingdom," The Economic Journal, 88, 661-692..

(23) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. 71. Deaton, A. (1992), "The Microeconomics and Macroeconomics of the Permanent Income Hypothesis," Temi di discussione, Numero 166. Flavin, M.A. (1981), "The Adjustment of Consumption to Changing Expectations about Future Income," Journal of Political Economy, 89, 974-1009.. e. d ~r. e. n. o. Hall, R E. (1978), "Stochastic Implications of the Life-Cycle-Permanent Income Hypothesis: Theory and Evidence," Journal of Political Economy, 86, 971-987. Hayashi, Fumio (1982), "The Permanent Income Hypothesis: Estimation and Testing by Instrumental Variables," Journal of Political Economy, 90, 895-916. Johansen, S. (1988), "Statistical Analysis of Cointegration Vectors," Journal of Economic Dynamics and Control, 12, 231-254.. d. Kitamura, Y. and P.c.ll. Phillips (1993), "Fully Modified IV, GIVE, and GMM Estimation. )f. with Possibly Non-Stationary Regressors and Instruments," Cowles Foundation Discussion. )f. Paper No. 1043, Yale University, May 1993.. e. Lee, Yuan-San and King-Lee Wang (1994), "Private Consumption Expediture in Taiwan: An. ~). Application of Error Correction Model," Essays on Economic Development of Taiwan. II. (Essays in Memory of Professor Hua len).. :-I n. Nelson, C.R (1987), ':A Reappraisal of Recent Tests of the Permanent Income Hypothesis," Journal of Political Economy, 95, 641-646. Nickell, Stephen (1985), "Error Correction, Partial Adjustment and All That: An Expository Note," Oxford Bulletin of Economics and Statistics, 47, 119-129. Park, J. Y. and P.c.B. Phillips (1988), "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, 4, 468-497. Park, J. Y. and P.c.B. Phillips (1989), "Statistical Inference in Regressions with Integrated. II. Processes: Part 2," Econometric Theory 5, 95-131. Phillips, P.c.B. and B. E. Hansen (1989), "Statistical Inference in Instrumental Variables. c. Regression with 1(1) Processes," Review of Economic Studies, 57, 99-125. Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit Root in Time Series Regression,". d. Biometrika, 75, 335-346. West, K.D. (1988), "The Insensitivity of Consumption to News about Income," Journal of. n r. Monetary Economics, 21, 17-33. Wu, Tsong-Min (1989), "Seasonal Fluctuations and Changes in Permanent Income," paper presented at the Conference of Business Cycles and Economic Growth of Taiwan.. ,f. Yang, Ya-Hwei (1980), "Permanent Income Theory," Master thesis, Department of Economics, National Taiwan University..

(24) 72. ACADEMIA ECONOMIC PAPERS 27 . 1. APPENDIX In this appendix, we explain the empirical implementation of FM-GMM and FM­ GIVE estimation and provide test statistics including those used to conduct instru­ ment validity test. The following is exerted from Kitamura.Y. and P.c.s. Phillips (1993) Section 7: A Practical Guide to Our Formulae for Empirical Implementation. Consider a multiple regression model:. Yt =. AXt. + UOt. where Yt is an n x 1 vector of dependent variables, A is an n x m coefficient matrix and. Xt. is an m-dimensional vector of cointegrated regressors.. Let Z denote a. q-vector of instruments. A.I FM·GMM Procedure Step 1: Run the naive IV regression. A = y' P:zX(X' p:zX)-l where P:z = Z(Z'Z)~rZ'. Then calculate the residual. Step 2: Use. UOt. obtained in Step 1 to calculate the distant matrix. 81'1'.. The spectral. estimator 8:z1' is defined as M~l. 8:z1' =. 2:. !uouo[1rk/MJ ® !uouo[-1rk/Mj. k=-M+1. where M. = o(T1/2). !ab(>-). = (1/271"). as T --+ T-1. 2:. j=-T+1. 00,. and! is the spectral density estimate of the form. wU/M)fabU)e ij )...

(25) 73. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. where •. T. rab(j) = (liT). 2:. at+jb~,. 1~t +j ~T. t=l. FM­. nstru­. lillips. and k is the bandwidth parameter and w(.) is the the lag window. Let. ation.. vec(A GMM ). (In ® X'Z)S:;,;,(In ® ZX')-l(In ® X' Z)S:;';'vec(yIZ). and calculate the GMM residual. natrix. lote a. UoGMMt. AGMMXt. Yt. Step 3: Use {UoGlvL'\1t} obtained in Step 2 and UOt = long run covariance matrices. (D.x~, D.z~)'. to estimate the. /11. nOa, naa and D.~oL:l.z = D.uoL:l.z - nOanaaD.uaL:l.z using kernel estimators. That is, T-l. ,. 2:. nab. w(jlk)ruaub(j). j=-T+l. T. )ectral. AUaUb . 2:. A. w(jlk)ruaub(j). j=O. are the kernel estimators used to construct the long run covariance matrix. Using the estimates. no. a,. naa and Li~oL:l.z obtained above, construct. vec(AGMM) = (In ® X'Z)S:;';'Un ® ZI X )-l(In ® X'Z)S:;';'vec(Y+'Z form where. ~. A. +. TD.uoL:l.z. ).

(26) ACADEMIA ECONOMIC PAPERS 27 : 1. 74. Calculate. witt UOGMMt. valil. Yt - AGMMXt. A.2. This completes the FM-GMM estimation. [We can iterate the process by returning to the beginning of Step 3 and using. {iiOGM Mt}. in place of. Firs. {UoGM Mt}.]. Step 4: Estimate ~uo~z [and no" again] using the GMM residual {iiOGMMtl obtained in Step 3 and call the estimate Auo~z [and noa). Calculate the corrected GMM residuals. whe:. oper. S. and its long run variance estimate. S. of C cons. [We could also calculate. SzT. again as in Step 2, but use. {UoGMMt}. in place of. {uot}·] Step 5: Construct the fully modified score matrix .::.+* =-2. -. S'. ­. = UOGMMZ. and the test statistics. Sl. these and.

(27) 75. TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. (+. =. (i + (2. (+ is used to test the validity of instruments, and is asymptotically. X2 distributed. with n(ij-m) degrees of freedom. The procedure is called the FM-GMM instrument validity test. A.2 FM·GIVE Procedure. First we assume the error term. is generated by the VAR(p) model. UOt. p. Assumption VR. UOt. L CrUOt-r + Et. = -. 1'=1. p. where. Et. i.i.d.(O,O"£). If C( L) = In. + L CrL r ,. where L denotes the backshift. 1'=1. operator, then the roots of IC(L)I = 0 are greater than one in absolute value. Step 1':. = Step. 1.. Step 2': Use UOt to estimate the VAR model in Assumption VR by the use of OLS. Using the estimates. C and. fe'. obtain the transformation matrix WT and. construct X* = WT(I (9 X), Z* = WT(I. (9. Z) vec(yl)*. = WTvec(Y'). Step 3': Construct the estimator vec(AGIV E) = (X*' Pz ·X*)-1 X*' Z* {(Z*'Z*)-1 Z*' vec(Y')*}. and associated residual. UOGIVEt = Yt - AOGIVEtXt. Step 4': Estimate Ooux,Ouxux and D. ux .6.z using {Uot} [or {UOGIVEtlJ and call these estimates OouxOuxux and Li ux .6.z. Construct the final FM-GIVE estimator.

(28) ACADEMIA ECONOMIC PAPERS 27 : 1 (1999). 76. vec(AG1VE). FM-GI 2 Xn(q-m. and calculate the residual. UoGIVEt. Yt - .AGIVEXt. [Once again we can iterate this process by returning to the beginning of Step 2' or Step 4' and using {uoGIVEd in place ofUot [{uoGIVEd.] Step 5/: Use {UOGIVEt} to calculate vec('&u,)~z')' Oou x and OUxu x ' Calculate the corrected GIVE residual. and its long run variance estimate. Step 6': Construct the fully modified GIVE score matrix ~#. #'­. ::'2GIVE = UOGIVEZ. and the test statistics. ~#. /. (2GIVE = veC(::'2GIVE) (Ooo.x ® Z Z). and. A. - / -. -1. ~#. vec(::'2GIVE).

(29) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. 77. FM-GIVE instrument validity tests are condueted using (6IV E as an asymptotic. Zl)}. or. he. ry. X;'(ij-m). criterion..

(30) ACADEMIA ECONOMIC PAPERS 27 : 1. 78. tlimpJTf~W1~Ef"JfiXJE FM-GMM ~~& FM-GIVE 1J1*Ef"J~m. *~.~.~~~mIA*.~*M~~'I~.ml~~4~**'fi. •. ~.At.n.~·A~~ml.~A~.ml~. •. ~~ml~.·#~~m. •• ~.~~-.~~~,~~.~ml.~~~*.~-(~*~) •• 11::. jldl~ • J!. fl *- 1i ' A11' A1Ujlf I ::t ~ it .11::. *: JlIJ ~ 1t IV, r ~ :rJi z ;fl:> it jlJ • ,j' , ~~:rJiA:~.~ml~#~~ml •• ~~~,lV,a~4ml~.~~~4~ *o •• ~ •• ~.~~~~~ml,~~~ •• t~ •• · . - ••• ~.* ~4~*~ •• ~~ ••• o.~ ~.A~.*M.~~~' •••• ~ ~. ~~·~ ••• lV,a •• ~~~~~,.*ml.~~~·~* •• ~ • • t~ft'~Aml~ •• ~~_~Al,~ ••• lV,a •• ~.~~~fiml ~.~~ •• ,*~~IV, •••• *-*~'~.~.Alo.~fi •• ~~At ~.~'~.fimIM •• ~~_~~~Al,lV,a~ ••• ~ •• ~~M~~ .ml~.~~ •• '~ ••••• ~~~Al,~~*.~*-.IV,.~ I~. .... •• ~ •• ~.~~~~MmIA.4ml • • ~'~~~MMW~.,~ft J!.~*aM.o~*,ml. •• ~~ •• :rJiAJ!.8.~~~~#.~·.~~#m. I A. ' A ~ ::F ft t*"* Johansen. ~. if *:. ,~. f't. t*. Phillips and Kitamura. (1993).

(31) TESTING THE PERMANENT INCOME HYPOTHESIS USING TAIWAN DATA. ~. FM-GIVE $> FM-GMM jj. *. 79. 0. ~rn.~~~'*~M1J.~M. •• Am.*~M •• M1J.a·.~, •. •• M1J.aJlj,.~,~~~m"A.~~*~.A.~*.*a ~.~:. ••. M1J·m"A.·~.~*M.~·.A*~ •• ·~~*~- • .::r:.JHl*' ~~*~-.tlJAfl;#*. 'I:!.. m i!. Jlj,. m­ rlf. 1J. .. ft -It.. fiE. m. 13).

(32)

數據

Updating...

參考文獻

相關主題 :