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Inflation, inflation uncertainty, and relative price variability.



1. Introduction

The early descriptive studies of the behavior of prices by Mills (1927) and Graham (1930) both found that the variability of relative price changes (henceforth From this time forward.

The term henceforth, when used in a legal document, statute, or other legal instrument, indicates that something will commence from the present time to the future, to the exclusion of the past.
, RPV RPV Remotely Piloted Vehicle
RPV Republican Party of Virginia
RPV Rancho Palos Verdes (California)
RPV Reactor Pressure Vessel
RPV Réseau Privé Virtuel (French: virtual private network) 
) increased with inflation.(1) A modem literature beginning with Vining Vining is the name of several places in the United States:
  • Vining in Iowa
  • Vining in Kansas
  • Vining in Minnesota
 and Elwertowski (1976) and Parks (1978) has re-examined this finding in a variety of settings and searched for the particular aspect of inflation that is most closely associated with RPV. Defining RPV as the sum of the squared deviations The definition of variance is either the expected value (when considering a theoretical distribution), or average (for actual experimental data) of squared deviations from the mean.  of the rates of change of various price subindexes from the rate of change of some overall price index, this literature has generally confirmed the basic relationship found by Mills and Graham.(2) However, there is little agreement regarding which particular aspect of inflation is most highly correlated with RPV. For instance, Parks finds that RPV increases primarily with the absolute value of unexpected inflation, while Tang tang, in zoology
tang: see butterfly fish.
 and Wang (1993) find that RPV increases with expected inflation as well as with the absolute value of unexpected inflation. At the same time, Fischer (1982) finds that RPV increases with expected inflation and positive instances of unexpected inflation but not with negative instances of unexpected inflation. Still others, such as Grier and Perry (1996), find that RPV increases only with ex ante inflation uncertainty.(3)

Expected inflation, realized unexpected inflation, and ex ante inflation uncertainty have all been proposed as determinants of RPV in various well-specified theoretical models as outlined in section 2. Despite this, no empirical analysis of the relationship between inflation and RPV has simultaneously included all of these aspects of inflation as explanatory variables. All of the empirical studies Empirical studies in social sciences are when the research ends are based on evidence and not just theory. This is done to comply with the scientific method that asserts the objective discovery of knowledge based on verifiable facts of evidence.  other than that of Grier and Perry (1996) use only measures of expected and unexpected inflation as explanatory variables. At the same time, while Grier and Perry extend the empirical literature by including a plausible measure of inflation uncertainty obtained from a generalized autoregressive conditional heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH)

A nonlinear stochastic process, where the variance is time-varying, and a function of the past variance. ARCH processes have frequency distributions which have high peaks at the mean and fat-tails, much like fractal distributions.
 (GARCH GARCH Generalized Autoregressive Conditional Heteroskedasticity ) model as an explanatory variable, they omit o·mit  
tr.v. o·mit·ted, o·mit·ting, o·mits
1. To fail to include or mention; leave out: omit a word.

2.
a. To pass over; neglect.

b.
 unexpected inflation from their model Specification.

This paper investigates the empirical relationship In science, an empirical relationship is one based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis.  between inflation and RPV in a model that incorporates measures of inflation uncertainty as well as expected and unexpected inflation. This allows a more complete test of the various competing theories of the inflation-RPV relationship. The finding is that no single theory explains the data and that, even in conjunction, the various proposed theories cannot explain the data.

The rest of the paper proceeds as follows. Section 2 reviews three theories that have been proposed as possible explanations of the relationship between RPV and various aspects of inflation. Section 3 discusses the producer price index data used in this paper. Section 4 describes the model of inflation that is used to generate measures of expected inflation, unexpected inflation, and inflation variability. Section 5 estimates a model that relates RPV to these measures of expected inflation, unexpected inflation, and inflation variability. Section 6 checks the robustness of the results by dividing the sample into halves and also by re-estimating the key regression with food and energy prices removed from consideration. Section 7 offers a brief conclusion.

2. Theories Linking RPV and Inflation

There are three well-developed theories that imply a relationship between RPV and inflation. These are (i) the signal-extraction model of Lucas (1972, 1973) and Barro (1976), (ii) the extension of the signal-extraction model by Hercowitz (1981) and Cukierman (1983), and (iii) the menu-costs model of Sheshinski and Weiss (1977) and Rotemberg (1983).(4) Each of these makes a distinct prediction regarding which aspect of inflation should be most closely related to RPV.

The Lucas-Barro signal-extraction model predicts that RPV should increase with ex ante inflation uncertainty. The greater is the ex ante variability of aggregate demand shocks (and ex ante inflation uncertainty), the more various real local shocks will be interpreted as aggregate shocks and will be responded to with price changes rather than quantity changes. Realized aggregate demand shocks, on the other hand, have no effect on RPV in the Lucas-Barro model because all firms respond identically to any given aggregate shock.

By contrast, realized aggregate demand shocks do affect RPV in the Hercowitz-Cukierman extension of the Lucas-Barro signal-extraction model in which price elasticities Price elasticities

The percentage change in quantity divided by a percentage change in the price. Answers the question: How much will the demand for my product decrease if I raise prices by 10%?
 of supply differ across firms. In the Hercowitz-Cukierman model, firms with high elasticities of supply adjust prices less in response to a given unexpected aggregate demand shock than do firms in markets with low elasticities of supply. Moreover, the magnitude of the discrepancy in price adjustments across sectors increases with the magnitude of the aggregate demand shock. This leads to the prediction that RPV will be associated with the magnitude of unexpected inflation, whether positive or negative.

Finally, anticipated changes in aggregate demand have no effect upon relative prices under either version of the signal-extraction model. Anticipated changes in aggregate demand simply result in a uniform increase or decrease in all nominal prices Nominal price

Price quotations on futures for a period in which no actual trading took place.
. By contrast, the menu-costs models of Sheshinski and Weiss (1977) and Rotemberg (1983) predict a positive relationship between RPV and expected inflation. These models predict that firms should use (S, s) pricing schemes if a fixed cost is incurred whenever output price is adjusted; that is, a firm should adjust its nominal price only when the real price of its good decays to its lower bound of s, at which time the price should be reset so that its real price equals the upper bound of S. Provided that firms do not adjust prices synchronously syn·chro·nous  
adj.
1. Occurring or existing at the same time. See Synonyms at contemporary.

2. Moving or operating at the same rate.

3.
a. Having identical periods.

b.
, these models predict that, as inflation increases, the difference between the optimal s and S will increase and more variability in relative price changes will be created. However, as noted by Danziger (1987), the period between observations on prices must be short compared to the period over which any given firm maintains a fixed nominal price for this prediction to hold. If, for instance, these two periods were equal in length, then RPV would always be zero despite (S, s) pricing policies and would be unrelated to expected inflation.

3. The Data

The data used are producer price index (PPI (1) (Pixels Per Inch) The measurement of the resolution of a monitor or scanner. For example, a monitor that is 16 inches wide and displays 1600 pixels across its width would have a resolution of 100 ppi (1600 divided by 16). ) data from 1948.01-1997.05 taken from the Bureau of Labor Statistics Bureau of Labor Statistics (BLS)

A research agency of the U.S. Department of Labor; it compiles statistics on hours of work, average hourly earnings, employment and unemployment, consumer prices and many other variables.
 web page.(5) Inflation in period t is measured as the difference in the logs of the PPI all-commodities price index in periods t and t - 1. The rate of price change for each of the two-digit-level subcomponents of the PPI all-commodities index is defined similarly. The variability of relative price changes in period t ([RPV.sub.t]) is measured as

[RPV.sub.t] = (1/n) [summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  of] [([[Pi].sub.it] - [[Pi].sub.t]).sup.2] where i=1 to n (1)

where [[Pi].sub.t] is the aggregate inflation rate and [[Pi].sub.it] is the rate of change of the ith subindex Subindex

A group of securities that is part of an index but is also tracked separately as a smaller, separate index.

Notes:
A software index, for example, would be a sub-index of a computer index.
. Weights are not available for the various subcomponents, so this is an unweighted index unweighted index

A stock price index that is calculated with equal weighting for each component. Unweighted indexes such as the Value Line averages are useful for individuals who invest an equal dollar amount in each stock.
 of RPV. Figures 1 and 2 depict de·pict  
tr.v. de·pict·ed, de·pict·ing, de·picts
1. To represent in a picture or sculpture.

2. To represent in words; describe. See Synonyms at represent.
 inflation and RPV, respectively, over the sample period.

RPV is measured using all of the 14 subcomponents of the PPI all-commodities price index for which data are available from 1948.01 onward on·ward  
adj.
Moving or tending forward.

adv. also on·wards
In a direction or toward a position that is ahead in space or time; forward.
.(6) This measure of RPV contrasts with the measures computed in Grier and Perry (1996) and Fischer (1981), which deliberately ignore the energy subcomponent sub·com·po·nent  
n.
A portion of a component, especially an electronic component; a subassembly.
 and the food and energy subcomponents, respectively, in an effort to control for supply shocks. Given the importance of market-specific shocks to the prediction of the Lucas-Barro model, the use of the most comprehensive measure of RPV possible seems appropriate in the present paper. At any rate, deleting the food and energy subcomponents from the measure of RPV does not qualitatively change the results, as shown in section 6.

4. A Model of Inflation and Inflation Variability

Preliminary investigation indicates that autoregressive moving average (ARMA) models of monthly inflation over the sample period are inadequate because the prediction errors of the best-fitting ARMA model, an ARMA ([1, 2, 3, 6], [1, 3, 12]) model, are heteroscedastic.(7) For this model, the Ljung-Box Q-statistic testing the serial independence of the squared residuals through lag 12 (Q[12] = 308.4) indicates rejection of the hypothesis of independence at the 1% level. Given this, inflation is modeled as a GARCH (1, 1) process to allow the conditional variance In statistics, conditional variance is a special form of the variance. If we have a conditional distribution Y|X the conditional variance is defined as



where
 of inflation to evolve with prediction errors regarding inflation. The following GARCH (1, 1) model of inflation over the 1948.01-1997.05 period minimizes the Akaike criterion among low-order GARCH models and has well-behaved disturbances.

[Mathematical Expression A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  Omitted] (2)

[Mathematical Expression Omitted]. (3)

(adjusted [R.sup.2] = 0.21)(8)

This model is estimated using the Marquardt algorithm. The t-statistics for the coefficient estimates are reported in parentheses See parenthesis.

parentheses - See left parenthesis, right parenthesis.
. All of the estimated coefficients in this model are statistically significant at the 5% level. The lag structure of this model of inflation is similar to that employed in Grier and Perry in their analysis of monthly PPI data.

As is common in this literature, expected and unexpected inflation are assumed to be generated by the process reported in Equation 2. The one-period-ahead forecast given by Equation 2 is taken as the measure of expected inflation (EI), and the difference between actual inflation and this forecast is taken as the measure of unexpected inflation (UI). Similarly, Equation 3 is assumed to generate a series for the conditional variance (CVARI) of inflation. By constructing these series in this fashion, it is being assumed that agents understand, at all times t, that Equations 2 and 3 describe the inflation process.

Finally, in order to test the prediction of the Hercowitz-Cukierman model that the sign of the inflation surprise should be irrelevant, two auxiliary series are created from UI. UIP UIP Usual interstitial pneumonia, see there  equals the absolute value of unexpected inflation when it takes positive values (and zero otherwise), while UIN UIN Unique Identification Number
UIN University Identification Number
UIN User Identification Number (ICQ)
UIN Quincy, IL, USA - Baldwin Field (Airport Code)
UIN Unit Identification Number
 is equal to the absolute value of unexpected inflation when it takes negative values (and zero otherwise). Actual inflation is denoted [[Pi].sub.t].

5. A Model of RPV

As a preliminary exercise, the positive relationship between inflation and RPV found by Parks and others is verified in the present data set. Regression of RPV on contemporaneous con·tem·po·ra·ne·ous  
adj.
Originating, existing, or happening during the same period of time: the contemporaneous reigns of two monarchs. See Synonyms at contemporary.
 inflation squared yields

[Mathematical Expression Omitted], (4)

(adjusted [R.sup.2] = 0.66)(9)

where the t-statistics in parentheses are based on standard errors computed according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 the procedure proposed in Newey and West (1987) to allow for residuals that exhibit both autocorrelation Autocorrelation

The correlation of a variable with itself over successive time intervals. Sometimes called serial correlation.
 and heteroscedasticity of unknown form. Clearly, the traditional positive association between inflation and RPV characterizes the data used here.

Note that here and throughout this paper explanatory variables are squared as needed as needed prn. See prn order.  so that the units of all the right-hand-side variables are the same and match that of RPV on the left-hand side left-hand side nizquierda

left-hand side left nlinke Seite f

left-hand side nlato or
. The qualitative results of this paper do not change if absolute values of the explanatory variables are used instead. Also, unless stated otherwise, the t-statistics reported below are based upon Newey-West standard errors due to the presence of heteroscedasticity and autocorrelation of unknown form in the error term.

The general model of RPV is given by Equation 5, in which RPV is regressed upon expected inflation (EI), positive unexpected inflation (UIP), negative unexpected inflation (UIN), and the conditional variance of inflation uncertainty (CVARI), all of which are generated by the inflation model of the previous section.(10) This regression yields

[Mathematical Expression Omitted], (5)

(adjusted [R.sup.2] = 0.72)(11)

where t-statistics are reported in parentheses. The estimates of the coefficients on expected inflation and positive unexpected inflation are positive and significant at the 1% level, while the estimate of the coefficient on the conditional variance of inflation is positive and significant at the 5% level. By contrast, the estimate of the coefficient on negative unexpected inflation is insignificant. Finally, the coefficient estimate on positive unexpected inflation is significantly larger than that on negative unexpected inflation. An F-test of the null hypothesis null hypothesis,
n theoretical assumption that a given therapy will have results not statistically different from another treatment.

null hypothesis,
n
 that the coefficients on positive unexpected inflation and negative unexpected inflation are equal yields an F(1, 581) statistic statistic,
n a value or number that describes a series of quantitative observations or measures; a value calculated from a sample.


statistic

a numerical value calculated from a number of observations in order to summarize them.
 of 26.2 and indicates rejection of equality at the 1% level. Thus, the Hercowitz-Cukierman version of the signal-extraction model, which predicts equal coefficients on positive and negative unexpected inflation, is rejected. The asymmetry Asymmetry

A lack of equivalence between two things, such as the unequal tax treatment of interest expense and dividend payments.
 found here between the coefficients on positive and negative unexpected inflation matches that reported in Fischer (1982).

One general conclusion that can be drawn is that no single one of the three models surveyed in section 2, by itself, predicts the pattern of coefficients found in Equation 5. Table 1 summarizes the predictions of each of the three theories as well as the empirical findings reported above.

Although the Lucas-Barro signal-extraction model and the menu-costs model both receive some support from the data, none of the middle three columns in Table 1 matches the final column, so the empirical results suggest rejecting the hypothesis that any of the three models, taken alone, completely explain the data.

This conclusion contrasts with that of Grier and Perry (1996) who, in a (narrower) test of [TABULAR tab·u·lar
adj.
1. Having a plane surface; flat.

2. Organized as a table or list.

3. Calculated by means of a table.



tabular

resembling a table.
 DATA FOR TABLE 1 OMITTED] the menu-costs model against the Lucas-Barro signal-extraction model, find that the Lucas-Barro model by itself explains the data. In an effort to reconcile these results, Equation 6 reports the results of a regression of RPV on inflation uncertainty and expected inflation. These are essentially the two explanatory variables utilized in the specification of Grier and Perry.(12) This specification yields

[Mathematical Expression Omitted], (6)

(adjusted [R.sup.2] - 0.15)(13)

where t-statistics are given in parentheses. The results in Equation 6 are somewhat similar to those found by Grier and Perry. Restricting the explanatory variables to this smaller set yields a coefficient estimate on inflation uncertainty that is positive and significant at all traditional levels, while the coefficient estimate on expected inflation, though positive, is significant only at the 10% level.(14)

A second conclusion that can be drawn from the results reported in Equation 5 is that all three models in conjunction cannot completely explain the data. The Hercowitz-Cukierman version of the signal-extraction model is the only one of the models that predicts a relationship between, inflation surprises and RPV. The fact that we reject the Hercowitz-Cukierman model despite finding a positive and significant coefficient on positive unexpected inflation suggests that our understanding of the relationship between inflation and RPV is incomplete. In fact, the results suggest that the most important part of the puzzle is missing because the estimate of the coefficient on positive unexpected inflation is the largest of the estimates.

6. Robustness

Two tests of the robustness of the findings of section 5 are performed. First, a Chow breakpoint The location in a program used to temporarily halt the program for testing and debugging. Lines of code in a source program are marked for breakpoints. When those instructions are about to be executed, the program stops, allowing the programmer to examine the status of the program  test is performed to determine whether behavior of RPV has changed over the 1948.01-1997.05 period. Second, the behavior of an alternative measure of RPV constructed without food and energy prices (some of the most volatile components of the PPI) is examined. In both cases, the essential conclusions remain unchanged.

Stability

A Chow breakpoint test of the specification in Equation 5 around the date 1972.12 does indicate a structural break in the behavior of RPV. The hypothesis that the coefficients are identical in the pre- and post-1972.12 periods yields an F(5, 576) statistic of 6.5, which indicates rejection of the null A character that is all 0 bits. Also written as "NUL," it is the first character in the ASCII and EBCDIC data codes. In hex, it displays and prints as 00; in decimal, it may appear as a single zero in a chart of codes, but displays and prints as a blank space.  at the 1% level. Equations 7 and 8 report the estimation results for the pre- and post-1972.12 periods, respectively as

[Mathematical Expression Omitted] (7)

(adjusted [R.sup.2] = 0.30)(15)

[Mathematical Expression Omitted]. (8)

(adjusted [R.sup.2] = 0.75)(16)

The estimation results for the post-1972.12 period reported in Equation 8 are essentially identical to those obtained in estimation using the full sample. The estimation results for the pre-1972.12 period reported in Equation 7, while different in some respects from those obtained in estimation using the full sample, reinforce the basic conclusions of section 5. The coefficient estimate on positive unexpected inflation in Equation 7 is again large and significant, and yet, the one model that purports to explain the relationship between inflation surprises and RPV, the Hercowitz-Cukierman model, is decidedly rejected in the pre-1972.12 period. An F-test of the equality of the coefficients on positive and negative unexpected inflation yields an F(1, 288) statistic of 7.4 and indicates rejection of equality at the 1% level. Moreover, the insignificant coefficient estimates on the conditional variance of inflation and expected inflation mean that the other two existing models, the Lucas-Barro model and the menu-costs model, do not even receive support as partial explanations of the data in the pre-1972 period.

An Alternative Measure of RPV

Several studies of inflation and RPV, including those of Fischer (1981) and Grier and Perry (1996) focus on measures of RPV that omit food and/or energy prices in an effort to control for supply shocks. Although the value of this is debatable de·bat·a·ble  
adj.
1. Being such that formal argument or discussion is possible.

2. Open to dispute; questionable.

3. In dispute, as land or territory claimed by more than one country.
 given the importance of local (sector-specific) shocks in generating the prediction of the Lucas-Barro model that RPV should increase with inflation uncertainty, the behavior of such a measure of RPV is considered below. The finding is that the behavior of RPV is little changed.

Eliminating food and energy prices means eliminating 3 of the 14 subcomponents of the PPI all-commodities index used above. These are farm products (wpu01), processed foods and feeds (wpu02), and fuels and related products and power (wpu05). Re-estimating the regression in Equation 5 with this new measure of RPV (labeled [RPV.sup.*]) yields

[Mathematical Expression Omitted], (9)

(adjusted [R.sup.2] = 0.69)(17)

where t-statistics are given in parentheses. The coefficient estimates and standard errors are similar to those reported in Equation 5. The coefficient estimate on negative unexpected inflation is insignificant, while the coefficient estimates on the other variables are positive and significant. One slight difference between the two sets of results is that here the coefficient on the conditional variance of inflation is significant only at the 10% level rather than at the 5% level. An F-test of the hypothesis that the coefficients on positive and negative unexpected inflation are equal yields an F(1, 581) statistic of 52.6. As above, this indicates rejection of the null at the 1% level and suggests the rejection of the Hercowitz-Cukierman model. Also, as above, the results support the menu-costs model and the Lucas-Barro model as partial explanations of the relationship between inflation and RPV. However, no single one of the three models fully explains the data nor can all of the models together do so.

7. Conclusion

Three existing models predict a relationship between the variability of relative price changes and some aspect of inflation. These are the menu-costs model, the Lucas-Barro signal-extraction model, and the Hercowitz-Cukierman extension of the Lucas-Barro model. The finding of this paper is that, taken separately or together, these models do not fully explain the U.S. data in the 1948.01-1997.05 period. First, the estimation results for the full sample as well as both the pre- and post-1972.12 subsamples indicate that we should reject the Hercowitz-Cukierman model, the only model of the three that predicts a positive relationship between RPV and the unexpected component of inflation. Yet, at the same time, the one coefficient estimate that is positive and significant in all of the samples is that on positive unexpected inflation. Second, while the estimation results for the full sample and the post-1972.12 subsample sub·sam·ple  
n.
A sample drawn from a larger sample.

tr.v. sub·sam·pled, sub·sam·pling, sub·sam·ples
To take a subsample from (a larger sample).
 provide some support for the menu-costs model and the Lucas-Barro model as partial explanations of the data, the results from the pre-1972.12 subsample do not even do that.

The fact that the relationship between RPV and positive unexpected inflation is completely different than that between RPV and negative unexpected inflation deserves further study and explanation. Models incorporating the assumption of downward rigidity rigidity /ri·gid·i·ty/ (ri-jid´i-te) inflexibility or stiffness.

clasp-knife rigidity
 in prices might possibly explain this asymmetry.(18) Models that take into account how money is injected in·ject·ed
adj.
1. Of or relating to a substance introduced into the body.

2. Of or relating to a blood vessel that is visibly distended with blood.



injected

1. introduced by injection.

2. congested.
 into and withdrawn from the economy might also be helpful in this regard.

I would like to thank Scott Atkinson, Kevin Grier, Bill Lastrapes, Art Snow, Ron Warren, and two anonymous referees for their helpful suggestions.

1 The variability of relative price changes is commonly referred to as relative price variability. This accounts for the acronym acronym: see abbreviation.


A word typically made up of the first letters of two or more words; for example, BASIC stands for "Beginners All purpose Symbolic Instruction Code.
 RPV. The phrase relative price variability is slightly misleading, however, because it suggests that it is dispersion dispersion, in chemistry
dispersion, in chemistry, mixture in which fine particles of one substance are scattered throughout another substance. A dispersion is classed as a suspension, colloid, or solution.
 in the levels of relative prices that is at issue in this literature rather than dispersion in the rates of change of relative prices.

2 The recent studies of Parsley parsley, Mediterranean aromatic herb (Petroselinum crispum or Apium petroselinum) of the carrot family, cultivated since the days of the Romans for its foliage, used in cookery as a seasoning and garnish.  (1996) and Debelle and Lamont (1997) have confirmed the positive relationship between inflation and RPV using city-level U.S. data.

3 Vining and Elwertowski (1976) also claim to find that RPV increases with inflation uncertainty (actually, inflation variability). However, as noted by Grier and Perry (1996), the evidence provided by Vining and Elwertowski supports only a link between RPV and the level of inflation.

4 Fischer (1981) surveys the various theories linking inflation and relative price variability.

5 The web address is http://stats.bls.gov.

6 These 14 subcomponents are farm products; processed foods and feeds; textile products and apparel; hides, skins, leathers, and related products; fuels and related products and power; chemical and allied products; rubber and plastic products; lumber lumber, term for timber that has been cut into boards for use as a building material. The major steps in producing lumber involve logging (the felling and preparation of timber for shipment to sawmills), sawing the logs into boards, grading the boards according to  and wood products; pulp paper and allied products; metal and metal products; machinery and equipment; furniture and household durables; nonmetallic non·me·tal·lic  
adj.
1. Not metallic.

2. Chemistry Of, relating to, or being a nonmetal.

Adj. 1.
 mineral products; and miscellaneous products. The 15th and final subcomponent, transportation equipment, is not included because data for that begins only in 1969.

7 This model is selected using the standard Box-Jenkins methodology. The correlogram for monthly inflation suggests the inclusion of AR(6) and MA(12) terms as well as an indeterminate That which is uncertain or not particularly designated.


INDETERMINATE. That which is uncertain or not particularly designated; as, if I sell you one hundred bushels of wheat, without stating what wheat. 1 Bouv. Inst. n. 950.
 number of low-order terms. The selected model minimizes the Akaike criterion among all ARMA models with any combination of low-order (three or less) AR and MA terms in addition to the AR(6) and MA(12) terms.

8 This model is also selected using the standard Box-Jenkins methodology. The correlogram for monthly inflation suggests the inclusion of AR(6) and MA(12) terms as well as an indeterminate number of low-order terms. The selected model minimizes the Akaike criterion among all GARCH models with any combination of low-order (three or less) AR and MA terms in addition to the AR(6) and MA(12) terms. For this model, the Q-statistic testing the serial independence in the residuals through 12 lags (Q[12] = 12.1) does not indicate rejection of the null hypothesis at the 5% level. Similarly, the Q-statistic testing the serial independence of the squared residuals through 12 lags (Q[12] = 15.5) does not indicate rejection of the null hypothesis at the 5% level (though it does at the 10% level).

9 Serial independence is rejected at the 1% level for the residuals (Q[12] = 104.9) as well as for the squared residuals (Q[121 = 260.3).

10 The use of the variables EI, UIN, UIP, and CVARI as regressors in Equation 5 is problematic. To the extent that these series measure true EI and true UIN and so forth with error, an errors-in-variables problem is created when conducting inference (logic) inference - The logical process by which new facts are derived from known facts by the application of inference rules.

See also symbolic inference, type inference.
 on the coefficients on true EI and true UIN and so forth. Pagan (1984) refers to this as the generated regressors problem. It is assumed here, as per common practice, that the estimates reported in Equation 5 and subsequently for EI, UIN, UIP, and CVARI are of practical interest because any such measurement errors are small.

11 Serial independence is rejected at the 1% level for the residuals (Q[12] = 58.0) as well as for the squared residuals (Q[12] = 191.0).

12 Grier and Perry actually use lagged inflation rather than expected inflation as the second explanatory variable. However, given the nature of the inflation model in Equation 2, these concepts are similar.

13 Serial independence is rejected at the 5% level for the residuals (Q[12] = 20.8). However, serial independence is not rejected for the squared residuals (Q[12] = 1.2).

14 It should be noted that Grier and Perry estimate inflation and RPV simultaneously. That approach is more efficient than the two-step approach of the present paper. The compensating benefit of the two-step approach is that it permits the straightforward use of EI, UIN, UIP, and CVARI in the model of RPV.

15 Serial independence is rejected at the 1% level for the residuals (Q[12] = 29.8). However, serial independence is not rejected at traditional levels of significance for the squared residuals (Q[12] = 10.1).

16 Serial independence is rejected at the 1% level for the residuals (Q[12] = 29.2) as well as for the squared residuals (Q[12] = 88.2).

17 Serial independence is rejected at the 1% level for the residuals (Q[12] = 45.2) as well as for the squared residuals (Q[12] = 44.7).

18 Ball and Mankiw (1994, 1995) propose models incorporating downward price rigidity that predict that inflation should increase with the variability of market-specific shocks.

References

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Ball, Laurence, and N. Gregory Mankiw. 1995. Relative-price changes as aggregate supply shocks. Quarterly Journal of Economics The Quarterly Journal of Economics, or QJE, is an economics journal published by the Massachusetts Institute of Technology and edited at Harvard University's Department of Economics. Its current editors are Robert J. Barro, Edward L. Glaeser and Lawrence F. Katz.  110:161-93.

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Extremely rapid or out of control inflation.

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