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‘On the formation of price expectations’ by König et al.
European Economic Review 16 (1981) 141-144. North-Holldnd Publishing Compa.ny
COMMENTS ‘On the Formation of Price Expectatiod
by Kihig et al.
H.R. WILLS London School c$ Economics and Political Science, Aldwych WC2A 2.4 E, UK
The paper by Konig, Nerlove and Oudiz is innovative in both the area examined and in the techniques used. In common with other new work it produces some surprise? rather than just confirming previous theories. My comments fall into two parts: the first is concerned with the technical aspects of the analysis, the second with the implications of the results. The main technique used by the authors is estimation of the log linear probability model. This is of course a model of association rather than of causation. As the categories are in this case ordered, the categories are scored and the probabilities represented by orthogonal polynomials. The score associated with each outcome is Score
Increase No change Decre:zse
(+ )
1
(0)
0
(I)
-1
(- )
For a one-way table with three groups the log linear model assumes that
log
(Pi
) = Oi
9
i= 1,2,3,
(2)
where tli is a linear function of unknown parameters. If the Ci is parameterised in terms of the score X, and orthogonal polynomials, then
where XI rather than being just the square of Xi is the component of the square of xi that is orthogonal to xi and the constant term. The advantage of using orthogonal polynomials is that the deletion of a high-order term does
H.R. Wills. Commenrs on the Ktinig et id. paper
the estimates of
the remaining low-order terms. The scoring (1) lies that the categories are in some sense equidistant or c quadra’tic term in (3) can either be interpreted as indicating atio~sl~ip is quadratic or alternatively that the categories are not
ing this framework to the 3 x 3 two-way table,
.2 +x,2mxtJ,
2 2 .2 tx2,xi.Vj+~22-~iJj,
(4)
ie interxtion
model. x, ,, the coefficient that is used as a primary association. is the linex interaction term. The Goodmana function of a!1 four of the high-order mteraction coefficients,
as remarked above the log linear probability model is a model of rather than of causation, fwhile most of the models presented in are causal. As has been pointed out by Nerlove and Press (1973), tor is believed to respontl to a second factor and the joint tion is log normal, then the distribution of the first factor conditional ond is logit in the same variables as the unconditional distribution. istribution is as in (4) it can be shown that
ere
(6) ever, some of the variables can be factored out. If this is done,
j=
g”‘J
/ I
c i
e”‘j
,
H.R. Wills, Cownents
on the Kiinig et al. paper
143
The terms that do not involve Xi factor out. Eq. (7) is most easily interpreted using the log odds property of the logistic model,
log (POjlP-
j)=wOj-w-
j
The fi on the second term in (9) and (10) is the result of norma!iLing the second-order orthogona!. polynomial. Thus the terms in thle first bracket measure the average effect of the polynomial in yi on both log odds ratios. The terms in the second bracket measure the degree to which the effect difIers for the two ratios. a,, measures the ‘average linear effect of yj on the two log odds ratios. The authors remark that in general estimating the parameters of the logit from the estimates of the log normal model will be more eficient. This is due to estimation of the log normal model requiring assumptions about the structure generating the complete unconditional distribution. lf the logit model is estimated directly no assumptions about the distribution of the ‘exogenous’ factors are required. If the assumptions about the s’tructure of the unconditional distribution are correct, the estimates obtained by estimating the full structure may be more efficient. If the assumptions are not true the estimates may be inconsistent, where the estimates from directly estimating thle conditional model would be consistent. The choice of approach depends on the quality of a priori information about the unconditional distribution available. It seems unlikely that such information will be available and that estimation of the conditional model will be safer for the cases where it is not. Examination of the results in this paper and of the complete results given in an earlier version indicates that both the quadratic parameter, a1 2 and the differential effect parameters, azl and az2 in eqs. (9) and (10) are often both numerically large and statistically significant. More seriously, many of the parameters are unstable over time, some randomly, some, particularly for the German data changing systematically. This suggests that the relationship between reality, the categories and the scoring varies over time or, to put it
’
H.R. Wills, Comments on the Kiinig et al. paper
a di~er~nt way, the model is unable to represent the actual process in a ous manner, which in turn suggests that it is only describing rather flying the structure. of variable which could generate this sort of effect is the general ation. If general inflation is low, there will be a certain rate of rice decrease/increase associated with the categories ( - ), and ( + ). In riod when inflation is higher, the average rate of price decrease with ( - ) will get smaller and the average rate of price increase ated with ( + ) will get larger. Thus the ‘c0ntent.s of our various * s will change. This in turn may be expected to lead to parameter ity. It may also mean that the higher rate of general inflation in relative tu Germany may be causing some of the reported differences ~r~~bl~!~here is that one suspects that the underlying structure is not e but continuous. Business men generate a qtlantitative expectation their change in prices. They then transform this quantitative nto a categorical response. It would be of. interest to know the uch a structure on the discrete estimators used here. One suspects me preliminary indications could be obt&ned fairly easily using rning now to the results, it seems to me that the most important result very strong association between price expectations and subsequent izations. This confirms what one would to some degree expect, that these are in fact plans. It is generally accepted that producers set in the short run and these results indicate that the short run extends ast two or three months. They show that the survey results are a forward indicator of price increases. n the other hand, it makes the logic of modelling prices independently of ut rather questionable. The results indica.te that both price plans and tions are serially correlated and it is not clear that the various simple tation models are capturing much more than this and the relation plans and realizations. odels of both output and prices for German data are -ng, despite their simplicity. Their failure for the French data is though it is possible that the relatively high rate of inflation in ans that the effect is swamped. s I said at the beginning, this is an innovative paper. It answers some s but raises a great many more. However, these questions are very ting and indicate that there is a great deal more work to do.
11J74, Univariate ar.d multivariate Corporation, Santa Monica, CA).
log hear
and logistic
models,
COMMENTS ‘On the Formation of Price Expectatiod
by Kihig et al.
H.R. WILLS London School c$ Economics and Political Science, Aldwych WC2A 2.4 E, UK
The paper by Konig, Nerlove and Oudiz is innovative in both the area examined and in the techniques used. In common with other new work it produces some surprise? rather than just confirming previous theories. My comments fall into two parts: the first is concerned with the technical aspects of the analysis, the second with the implications of the results. The main technique used by the authors is estimation of the log linear probability model. This is of course a model of association rather than of causation. As the categories are in this case ordered, the categories are scored and the probabilities represented by orthogonal polynomials. The score associated with each outcome is Score
Increase No change Decre:zse
(+ )
1
(0)
0
(I)
-1
(- )
For a one-way table with three groups the log linear model assumes that
log
(Pi
) = Oi
9
i= 1,2,3,
(2)
where tli is a linear function of unknown parameters. If the Ci is parameterised in terms of the score X, and orthogonal polynomials, then
where XI rather than being just the square of Xi is the component of the square of xi that is orthogonal to xi and the constant term. The advantage of using orthogonal polynomials is that the deletion of a high-order term does
H.R. Wills. Commenrs on the Ktinig et id. paper
the estimates of
the remaining low-order terms. The scoring (1) lies that the categories are in some sense equidistant or c quadra’tic term in (3) can either be interpreted as indicating atio~sl~ip is quadratic or alternatively that the categories are not
ing this framework to the 3 x 3 two-way table,
.2 +x,2mxtJ,
2 2 .2 tx2,xi.Vj+~22-~iJj,
(4)
ie interxtion
model. x, ,, the coefficient that is used as a primary association. is the linex interaction term. The Goodmana function of a!1 four of the high-order mteraction coefficients,
as remarked above the log linear probability model is a model of rather than of causation, fwhile most of the models presented in are causal. As has been pointed out by Nerlove and Press (1973), tor is believed to respontl to a second factor and the joint tion is log normal, then the distribution of the first factor conditional ond is logit in the same variables as the unconditional distribution. istribution is as in (4) it can be shown that
ere
(6) ever, some of the variables can be factored out. If this is done,
j=
g”‘J
/ I
c i
e”‘j
,
H.R. Wills, Cownents
on the Kiinig et al. paper
143
The terms that do not involve Xi factor out. Eq. (7) is most easily interpreted using the log odds property of the logistic model,
log (POjlP-
j)=wOj-w-
j
The fi on the second term in (9) and (10) is the result of norma!iLing the second-order orthogona!. polynomial. Thus the terms in thle first bracket measure the average effect of the polynomial in yi on both log odds ratios. The terms in the second bracket measure the degree to which the effect difIers for the two ratios. a,, measures the ‘average linear effect of yj on the two log odds ratios. The authors remark that in general estimating the parameters of the logit from the estimates of the log normal model will be more eficient. This is due to estimation of the log normal model requiring assumptions about the structure generating the complete unconditional distribution. lf the logit model is estimated directly no assumptions about the distribution of the ‘exogenous’ factors are required. If the assumptions about the s’tructure of the unconditional distribution are correct, the estimates obtained by estimating the full structure may be more efficient. If the assumptions are not true the estimates may be inconsistent, where the estimates from directly estimating thle conditional model would be consistent. The choice of approach depends on the quality of a priori information about the unconditional distribution available. It seems unlikely that such information will be available and that estimation of the conditional model will be safer for the cases where it is not. Examination of the results in this paper and of the complete results given in an earlier version indicates that both the quadratic parameter, a1 2 and the differential effect parameters, azl and az2 in eqs. (9) and (10) are often both numerically large and statistically significant. More seriously, many of the parameters are unstable over time, some randomly, some, particularly for the German data changing systematically. This suggests that the relationship between reality, the categories and the scoring varies over time or, to put it
’
H.R. Wills, Comments on the Kiinig et al. paper
a di~er~nt way, the model is unable to represent the actual process in a ous manner, which in turn suggests that it is only describing rather flying the structure. of variable which could generate this sort of effect is the general ation. If general inflation is low, there will be a certain rate of rice decrease/increase associated with the categories ( - ), and ( + ). In riod when inflation is higher, the average rate of price decrease with ( - ) will get smaller and the average rate of price increase ated with ( + ) will get larger. Thus the ‘c0ntent.s of our various * s will change. This in turn may be expected to lead to parameter ity. It may also mean that the higher rate of general inflation in relative tu Germany may be causing some of the reported differences ~r~~bl~!~here is that one suspects that the underlying structure is not e but continuous. Business men generate a qtlantitative expectation their change in prices. They then transform this quantitative nto a categorical response. It would be of. interest to know the uch a structure on the discrete estimators used here. One suspects me preliminary indications could be obt&ned fairly easily using rning now to the results, it seems to me that the most important result very strong association between price expectations and subsequent izations. This confirms what one would to some degree expect, that these are in fact plans. It is generally accepted that producers set in the short run and these results indicate that the short run extends ast two or three months. They show that the survey results are a forward indicator of price increases. n the other hand, it makes the logic of modelling prices independently of ut rather questionable. The results indica.te that both price plans and tions are serially correlated and it is not clear that the various simple tation models are capturing much more than this and the relation plans and realizations. odels of both output and prices for German data are -ng, despite their simplicity. Their failure for the French data is though it is possible that the relatively high rate of inflation in ans that the effect is swamped. s I said at the beginning, this is an innovative paper. It answers some s but raises a great many more. However, these questions are very ting and indicate that there is a great deal more work to do.
11J74, Univariate ar.d multivariate Corporation, Santa Monica, CA).
log hear
and logistic
models,