Formations of expectations in
econometric models
Gregory C Chow
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.
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:
• Ct = 24106.1(17986.2) + .641(.0892) Ct-1 + .2756(.0621) Y*t
R2= 0.9992; s = 88650
(2)
Macro-economic model for China
JPE(1985), Econ Letters(2010)
• 1985 paper, 1952-1982. 2010 paper, data 1978-2006:
• (2a) Ct = 218.86 + 1.067(.074) Ct-1 – 0.0051(.0371)Y*t R2=
0.9985; s = 271.24
• This result confirms the permanent income hypothesis of Hall
perfectly. Re-estimated the consumption function to obtain
• (2) Ct = 226.05(91.78) + 1.0570(.0079) Ct-1 R2 = 0.9985; s= 266
• The investment function is
• (3a) It = -399.04(139.79) + 2.4149(.6470) Y*t – 2.2861(.6281)
Yt-1 + .2233(.2369) It-1
R2 =.9968;
s= 327.4
• (3) It = -186.23(120.84) + 1.7782(.6513)(Y*t-Yt-1)
+.6866(.1589) It-1
R2 = .9960; s = 359.28
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 Yt-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.
Taiwan consumption function supports
Friedman’s permanent income
•
•
•
•
•
•
•
•
The consumption function of Friedman (1957) states C = a Yp :
By adaptive expectations, Yp -Yp(t-1) = b[Y(t) - Yp(t-1)], implying
Yp = bY(t) +(1-b)Yp(t-1) = bY(t) + b(1-b)Y(t-1) + b(1-b)2Y(t-2)+ ...
Under adaptive expectations permanent income is a weighted
average of current income Y(t) and permanent income Yp(t-1) of the
preceding period with weights b and (1-b) respectively. By repeated
substitutions of lagged Y’s for lagged Yp backward in time Yp equals
to the right-hand side of the above equation. When this expression
is substituted into consumption function we obtain
Ct = a [bYt + b(1-b)Yt-1 + b(1-b)2Yt-2 + ...]
Ct-1 = a [bYt-1 + b(1-b)Yt-2 + b(1-b)2Yt-3 + ...]
which imply
Ct = abYt + (1-b)Ct-1
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 Ct-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?
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 Yp
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.
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)
R2= 0.1581 s = .02901
(7) Taiwan 1951-2010
• ΔlogY(t) = 0.3313(.1290) ΔlogY(t-1) + 0.0506(.0146)
R2= 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.
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)?
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.
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 pt at
the end of year t and dividend dt distributed during year t for blue chip
stocks in Taiwan to perform the following regression
• (3) log pt = 2.610(0.075) + 0.281(0.089) log dt + 0.414(0.098)[log dt
- log dt-3]
• Both explanatory variables are statistically significant. There were 445
observations covering years from 1971 to 2010.
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)
pt = bEt(pt+1 + dt).
• Stock price pt at the beginning of year t equals discounted
expected sum of stock price pt+1 at the beginning of year t+1
and dividend dt paid during year t.
Why rational expectations is rejected
in testing present value models
• The expectation here means the subjective expectation of investors who
are willing to pay pt now because they think that a year from now the
stock price pt+1 and dividend dt will be such that their discounted sum will
equal the current price pt.
• 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 dt 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 pt.
• Chow (1989) provides strong evidence showing the discrepancy between
pt 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.
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.
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
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.
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.
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).
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.
•
Thank you