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Formations of expectations in econometric models Gregory C Chow

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Importance of expectations in empirical work: national income determination in China Chow (1985, JPE; 2001, Econ Letters) estimated consumption function based on Hall (1978) and investment function based on the accelerations principle using data for China. 1. Hall’s consumption function (random walk with drift) fits data for China but not for Taiwan, with Y* estimated by 2SLS significant in the Taiwan consumption function. 2. I discovered that the Taiwan data support the permanent income hypothesis of Friedman (1957) based on adaptive expectations. 3. I found an explanation why two sets of data support two different hypotheses on expectations. Robert Lucas and Thomas Sargent received Nobel Prize for work on rational expectations.

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Consumption function of the econometric model – Taiwan data The three structural equations include (1) the national income identity Y = C + I + X, with Y, C, I and X denoting real GDP, consumption, investment and exports minus imports respectively; (2) a consumption function linear in C(t-1) and Y and (3) an investment function linear in Y, Y(t-1) and I(t-1). The endogenous variables are C, I and Y; the predetermined variables are X, Y(t-1), C(t-1) and I(t-1). In the first stage, Y* is estimated by regressing Y on the predetermined variables using 60 annual observations from 1951 to 2010. Y* is significant in the consumption function: C t = 24106.1(17986.2) +.641(.0892) C t-1 +.2756(.0621) Y* t R 2 = 0.9992; s = 88650 (2)

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Macro-economic model for China JPE(1985), Econ Letters(2010) 1985 paper, 1952-1982. 2010 paper, data 1978-2006: (2a) C t = 218.86 + 1.067(.074) C t-1 – 0.0051(.0371)Y* t R 2 = 0.9985; s = 271.24 This result confirms the permanent income hypothesis of Hall perfectly. Re-estimated the consumption function to obtain (2) C t = 226.05(91.78) + 1.0570(.0079) C t-1 R 2 = 0.9985; s= 266 The investment function is (3a) I t = -399.04(139.79) + 2.4149(.6470) Y* t – 2.2861(.6281) Y t-1 +.2233(.2369) I t-1 R 2 =.9968; s= 327.4 (3) I t = -186.23(120.84) + 1.7782(.6513)(Y* t -Y t-1 ) +.6866(.1589) I t-1 R 2 =.9960; s = 359.28

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Same model valid for 1952-1982 and for 1978-2006 In Chow (1985) I reported results similar to equations (2) and (3) obtained by using Chinese annual data from 1952 to 1982. In the consumption function the coefficient of lagged consumption was almost equal to one and the coefficient of income was zero. In the investment equation the coefficient of Y t-1 was negative and slightly less in magnitude than the coefficient of Y and I replaced these variables by their difference as in equation (3). The coefficient of this difference in the investment equation was smaller than 1.7782 possibly because the ratio a of capital stock to output was smaller and the adjustment coefficient b for capital stock to reach equilibrium was also smaller before 1978. In conclusion I have found that the permanent income hypothesis of Hall (1978) to explain consumption and the accelerations principle to explain investment are well supported by Chinese macro data for the periods 1952-1982 and 1978-2006 as well.

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Taiwan consumption function supports Friedman’s permanent income The consumption function of Friedman (1957) states C = a Y p : By adaptive expectations, Y p -Y p (t-1) = b[Y(t) - Y p (t-1)], implying Y p = bY(t) +(1-b)Y p (t-1) = bY(t) + b(1-b)Y(t-1) + b(1-b) 2 Y(t-2)+... Under adaptive expectations permanent income is a weighted average of current income Y(t) and permanent income Y p (t-1) of the preceding period with weights b and (1-b) respectively. By repeated substitutions of lagged Y’s for lagged Y p backward in time Y p equals to the right-hand side of the above equation. When this expression is substituted into consumption function we obtain C t = a [bY t + b(1-b)Y t-1 + b(1-b) 2 Y t-2 +...] C t-1 = a [bY t-1 + b(1-b)Y t-2 + b(1-b) 2 Y t-3 +...] which imply C t = abY t + (1-b)C t-1

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Taiwan data support Friedman; China mainland data support Hall From our estimated equation, ab =.2756 1-b =.6410 or b =.3590 and a =.2756/.3590 =.7677. The estimate.7677 for a, the fraction of national income devoted to consumption, is reasonable. According to the permanent income hypothesis of Hall (1978), the coefficient of C t-1 equals 1 and the coefficient of Y* should be zero. This hypothesis was confirmed by Chow (1985, 2010, 2011) using data for China. Why do the data for Taiwan and for China support different versions of the permanent income hypothesis?

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Difference in behavior between Taiwan’s and China’s consumers For the Taiwan consumers to use a weighted mean to forecast expected income with geometrically declining weights, or to give more weights to more recent income, they must believe that the changes in income are serially correlated. If the changes in income Y are statistically independent the appropriate mean is an unweighted mean of past incomes, leading to no change in the estimate of expected income Y p from last year, or to using permanent income last year or the consumption of last year to predict C(t) as suggested by the Hall consumption function.

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Testing whether Taiwan data justify the formation of expectations by Taiwan consumers To find out whether movement of past Y(t-k) in Taiwan did affect Y(t), more so than movement of past Y(t-k) affected Y(t) in China, I perform a regression of ΔlogY(t) on ΔlogY(t-1) for both economies. ΔlogY(t) = 0.3953(.1219) ΔlogY(t-1) + 0.0431(.0095) R 2 = 0.1581 s =.02901 (7) Taiwan 1951-2010 ΔlogY(t) = 0.3313(.1290) ΔlogY(t-1) + 0.0506(.0146) R 2 = 0.1106 s =.07955 (8) China 1952-2008 The relative magnitudes of these two standard errors confirm our explanation as to why data for Taiwan support Friedman’s version of the permanent income hypothesis and data for China support the Hall version.

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Question for further research Changes in real income in Taiwan have been more predictable than in China, leading the Taiwan consumers to use current income to estimate permanent income as specified by the Friedman theory of permanent income to a larger extent than consumers in China. The next question is to find the reasons for the difference in the dynamics between the time series data on national income of Taiwan and China. What economic history of the two economies can explain the difference between regression equations (7) and (8)?

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Usefulness of adaptive v. rational expectations in economics 1. Adaptive expectations is a useful hypothesis on human behavior in forming expectations. See evidence from paper 1, Chow (1989, REStat), Chow (2007, chap 14). I also present a methodological argument for using adaptive expectations in paper 2. 2. State reasons why the rational expectations hypothesis was endorsed by the economic profession without sufficient evidence. 3. Both hypotheses have their places and can be used effectively.

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Why adaptive expectations is a reasonable hypothesis Hypothesis A: Economic agents form expectations of an economic variable by taking a weighted mean of its past values. Adaptive Expectations Hypothesis: As a special case of Hypothesis A, a mean with geometrically declining weights is taken, as shown before. I will cite strong evidence for the empirical validity of Hypothesis A. Once Hypothesis A is accepted, I will cite additional evidence supporting the Adaptive Expectations Hypothesis. As evidence supporting Hypothesis A, I have used data on the price p t at the end of year t and dividend d t distributed during year t for blue chip stocks in Taiwan to perform the following regression (3) log p t = 2.610(0.075) + 0.281(0.089) log d t + 0.414(0.098)[log d t - log d t-3 ] Both explanatory variables are statistically significant. There were 445 observations covering years from 1971 to 2010.

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Chow (REStat 89) provides strong evidence for adaptive expectations Chow (1989) provided very strong econometric evidence supporting the adaptive expectations hypothesis against the rational expectations hypothesis for the present-value model. This model was applied to explain stock price as a discounted sum of expected future dividends and to explain long term interest rate as a sum of expected future short-term rates. Let me explain why the hypothesis of rational expectations is strongly rejected by the data. An implication of the present value model of stock price is (4) p t = bE t (p t+1 + d t ). Stock price p t at the beginning of year t equals discounted expected sum of stock price p t+1 at the beginning of year t+1 and dividend d t paid during year t.

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Why rational expectations is rejected in testing present value models The expectation here means the subjective expectation of investors who are willing to pay p t now because they think that a year from now the stock price p t+1 and dividend d t will be such that their discounted sum will equal the current price p t. The econometrician believing in the hypothesis of rational expectations is required to have an econometric model to forecast the future p t+1 and future d t as yet to be distributed during year t. There is no reason to believe that the expected values so estimated will have a sum, after discounting, that equals the actual p t. Chow (1989) provides strong evidence showing the discrepancy between p t and its estimate by rational expectations. The hypothesis of rational expectations is wrong whenever the econometrician’s model is poor. This happens often. Why not save the expectations part of the model by assuming rational expectations.

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Why a skeptic of adaptive expectations has a difficult task to justify his view If economic agents use past trend to project into the future (to form expectations of the future), a skeptic of adaptive expectations has to present strong evidence that the past trend cannot be projected by using geometrically declining weights as stated by the adaptive expectations hypothesis. The task for the skeptic is to reject the null hypothesis of using a set of geometrically declining weights to estimate the expected variable in question. He may be able to show in a few econometric studies that some other weighting scheme is econometrically better. But if he rejects the adaptive expectations hypothesis his task is to show that in empirical studies yet to be performed the use of geometrically declining weights would be statistically rejected. Evidence in Chow (1989) and Chow (2007) supporting the use of geometrically declining weights as specified by the adaptive expectations hypothesis makes the task of the skeptic difficult.

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The Lucas critique (1976) The popularity of the rational expectations hypothesis began with the critique of Lucas (1976) which claimed that existing macro econometric models of the time could not be used to evaluate effects of economic policy because the parameters of these econometric models would change when the government decision rule changed. A government decision rule is a part of the environment facing economic agents. When the rule changes, the environment changes and the behavior of economic agents who respond to the environment changes. Economists may disagree on the empirical relevance of this claim, e.g., by how much the parameters will change and to what extent government policies can be assumed to be decision rules rather than exogenous changes of a policy variable. I accept the Lucas critique to continue discussion.

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Logical mistake of many economists Assuming the Lucas critique to be valid, economists can build structural econometric models with structural parameters unchanged when a policy rule changes. Such a solution can be achieved by assuming rational expectations, together with some other modeling assumptions. I also accept this solution. In the late 1970s, the economics profession (1) accepted the Lucas critique, (2) accepted the solution to the Lucas critique in which rational expectations is used and (3) rejected the adaptive expectations hypothesis possibly because the solution in (2) required the acceptance of the rational expectations hypothesis. Step (3) is not justified by (1) and (2) because rational expectations can be empirically incorrect.

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Insufficient empirical evidence supporting rational expectations and rejecting adaptive expectations There was insufficient evidence supporting the hypothesis of rational expectations when it was embraced by the economic profession in late 1970s. Lucas and Sargent, 1981. Rational Expectations and Econometric Practice. University of Minnesota Press, contains articles on how to apply rational expectations models once you decide to use them but little evidence supporting the hypothesis. Being interested in the topic I contributed two papers in these two volumes.

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Recommendations to economists For the purpose of finding good estimates of psychological expectations as required in the study of economic behavior, adaptive expectations should be used whenever the economist believes that the economic agents in question form psychological expectations by taking a mean of past values with geometrically declining weights. He should assume rational expectations if he believes that his econometric model can generate mathematical expectations that are closer to the psychological expectations of the economic agents than the assumption of adaptive expectations can. It would also be of interest for the economist to compare the two expectations hypotheses as was done in Chow (1989).

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Role of rational expectations in econometric models An economist using rational expectations will use the mathematical expectations derived from his econometric model as psychological expectations of the economic actors in the model. Rational expectations are good if the model formulated by the economist is good and the economic actors actually follow the model in formulating their psychological expectations. In the construction of econometric models economists need to have a correct hypothesis on the formation of expectations by economic agents.

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