This section examines the relationship between the rate of inflation and other macroeconomic variables in Mexico over the period 1988–91, using a vector autoregressions model.

During this period, the sustained implementation of comprehensive adjustment policies reduced sharply the rate of inflation in Mexico. Those policies have included fiscal and monetary restraint, trade reforms, and the determination of certain key prices in the framework of an agreement among the Government, labor, and business firms, namely the PACTO (Pacto de Solidaridad Económica), and its successor, the Pece (Pacto para la Estabilidad y el Crecimiento Económico). Within this framework, a social consensus has been formed on the target path for the exchange rate, the minimum wage, public sector prices, and some private sector prices. These targets have been revised periodically in light of the evolution of the economy.

This section examines to what extent the rate of inflation was associated with the evolution of other macro-economic variables, such as the rate of change of the exchange rate, foreign inflation, wage developments, and monetary expansion, during this period of reduced inflation. The vector autoregression methodology used here is based on the estimation of reduced form equations and does not require being explicit about the specific channels of transmission of shocks in the economy. Therefore, the results presented do not constitute a test of any specific theory about inflation. Instead, they summarize key stylized facts about inflation dynamics and describe the statistical association between a number of related macroeconomic variables. They are also useful for testing some selected hypotheses about the determinants of inflation and for forecasting.^{1}

The estimation results indicate that, during the period under study, the rate of inflation responded significantly to adjustments in public sector prices, foreign inflation, exchange rate changes, and monetary expansion, and to a lesser extent to wage changes. The results also reveal some puzzling aspects of the behavior of wages.

The remainder of the section is organized as follows. The first part describes briefly the methodology. The second presents the variables included in the model, the estimation results, some exercises regarding the dynamic response of inflation to various shocks, and some fore-casting exercises. The third part contains a summary of the results and some concluding remarks. An appendix contains a more formal presentation of the methodology.

## Methodology

The vector autoregression (VAR) methodology consists of estimating a system of equations in which each endogenous variable is regressed on lagged values of all endogenous variables and on current (and sometimes also lagged) values of the exogenous variables. The resulting system can be interpreted as the reduced form equations of some underlying system of structural equations.

These reduced form equations capture the effects of the exogenous variables and the intertemporal relationship between the endogenous variables in the model. The contemporaneous relationship between the endogenous variables, on the other hand, is estimated from the residuals of the regressions, under a suitable set of assumptions regarding which endogenous variables are more likely to adjust to changes in other endogenous variables within the same period. This permits the identification of the exogenous shocks, or innovations, that affect the endogenous variables in the structural system. In the interpretation of this type of model, the dynamics of the endogenous variables are driven by the realization of the innovations and the evolution of the exogenous variables. The innovations represent shocks that are not captured by the exogenous variables explicitly incorporated into the model.

This methodology is useful for tracing the effects of the various innovations on the endogenous variables. The procedure to follow is (a) estimate the reduced form system; (b) use the estimated reduced form errors together with some additional assumptions to identify the innovations; and (c) use those results to derive the dynamic effect of the innovations on the path of the endogenous variables. The model also generates forecasts for the endogenous variables, for a given path of the exogenous variables, by assuming that there are no innovations during the forecasting period. The variance of the forecast error can be decomposed into the parts stemming from the uncertainty regarding the evolution of each type of innovation. The model can also be used to examine the role played by the various innovations in a particular period, within the estimation period, by means of a historical decomposition of the forecast error. For this, the model is used to forecast within sample, and the actual forecast error is decomposed into the parts due to the accumulated effects of the various innovations.

## A Model for Mexican Inflation

### Variables Included in the Model

The rate of inflation in Mexico is postulated to depend on a number of factors representing cost and demand pressures. (1) One factor is imported inflation, defined as the nominal effective rate of depreciation of the Mexican peso plus the rate of foreign inflation. Imported inflation may affect the domestic price level directly through its effect on the prices of traded consumer goods, and indirectly through its effect on the costs of producing non-traded goods that use traded inputs. (2) Another factor affecting domestic inflation is the behavior of wages, through its effect on costs of production. (3) The evolution of minimum wages may also affect inflation to the extent that prices react immediately to anticipated wage increases that may be influenced by the level of the minimum wage. (4) Domestic inflation is also likely to be affected by the rate of monetary expansion, under the assumption that higher monetary expansion will be associated with higher demand pressures. (5) Another variable that needs to be included is the adjustment of public sector prices. Since detailed information on those adjustments is not available, and since most of the important revisions of public sector prices took place in the months of December and January, an attempt is made to capture their effect by using dummy variables for those two months.

The variables included in the model (described more fully below) are denoted by:

p | rate of domestic inflation |

ii | rate of imported inflation |

w | rate of wage increase |

z | rate of minimum wage increase |

m | rate of monetary expansion |

s1 | dummy variable for January |

s12 | dummy variable for December. |

p | rate of domestic inflation |

ii | rate of imported inflation |

w | rate of wage increase |

z | rate of minimum wage increase |

m | rate of monetary expansion |

s1 | dummy variable for January |

s12 | dummy variable for December. |

The exogenous variables in the system include policy variables and external factors. Thus, they include the dummies for January and December, the rate of change of minimum wages, and the rate of imported inflation. Imported inflation is exogenous because the foreign rate of inflation is exogenous, and the nominal rate of crawl of the Mexican peso with respect to the U.S. dollar was predetermined during the period under consideration.^{2}

In addition to the rate of inflation, the other endogenous variables in the system are the rate of increase in nominal wages and the rate of monetary expansion. Nominal wages are endogenous because they are likely to respond to changes in inflation, and to the external and policy factors mentioned above. Unless money is represented by the monetary base, monetary expansion is endogenous to the extent that the multiplier responds to other endogenous variables in the system. Under a regime of a predetermined exchange rate, the monetary base also becomes endogenous since changes in domestic credit can be partially or totally offset by endogenous changes in international reserves. Furthermore, monetary expansion may also become endogenous because the authorities adjust domestic credit in response to the evolution of other endogenous variables in the economy.

The estimation of the system thus described would include one regression for each of the endogenous variables: the rate of inflation (p), the rate of growth of nominal wages (w), and the rate of monetary expansion *(m)*. The regressors would include lagged values of all endogenous variables, and current (and maybe also lagged) values of the rate of growth of minimum wages (z), the dummy variables for January and December (s1 and s12), and the rate of imported inflation *(if)*. In order to consider explicitly the dynamics of imported inflation as one of the sources of errors in forecasting domestic inflation, the model was modified, however. An additional equation was estimated for imported inflation as a function of its own lagged values, and the errors from this regression were considered as innovations in imported inflation. In addition, the current value of imported inflation was excluded from the estimated equations for the endogenous variables.

With this modification, the estimated model consists of four equations: *ii* is regressed on its own lagged values; and *p, w*, and *m* are regressed on lagged values of p, w, m, and *ii*, and on *z, s*1, and *s*12. In the interpretation of this model, the evolution of imported inflation depends only on innovations in *ii*, while the evolution of domestic inflation, money expansion, and wage increases depend on innovations in *ii, p, m, w*, and on the values of *z, s1*, and *s*12.

### Data and Estimation Period

The model was estimated with monthly data for the period June 1988-March 1991. The beginning of the period was chosen so as to exclude the initial few months with the new policy regime in which the structure of the system was in transition from a period of high inflation to a period of low inflation. The end of the period was determined by the availability of data on wages.

Data on prices, wages, and money were obtained from *Indicadores Econímicos* (Banco de Mexico). The rate of inflation (*p*) was measured by the monthly rate of increase of the national consumer price index.^{3} The rate of monetary expansion (*m*) was measured by the monthly rate of growth of M2.^{4} As explained below, this variable was seasonally adjusted before the estimation. The monthly rate of growth of wages (*w*) was derived from the series on average hourly salaries and benefits in the manufacturing sector. Since this series includes the payment of a large annual bonus in December of each year, the rate of growth of wages presents a sharp increase every December and a sharp decline every January. In order to correct for this factor, and in the absence of accurate information about the size of the annual bonus, the value of wages for December was estimated by taking the average values of the two adjacent months. Imported inflation (*ii*) was calculated as the sum of the nominal effective rate of depreciation of the Mexican peso with respect to the currencies of a group of countries plus the weighted average of the rates of inflation of those countries, with data from *International Financial Statistics*.^{5} The rate of increase of minimum wages (*z*) was measured on the basis of the national average minimum wage as reported by the Comisión Nacional de los Salarios Minimos.

Some characteristics of the variables included in the analysis are presented in Table 10 and Chart 18. The means of the monthly growth rates of wages and money are substantially higher than the mean of domestic inflation, reflecting the increase in real wages and the process of monetization of the economy in this period of reduced inflation. Minimum wages, on the other hand, increased by less than domestic prices. The mean of imported inflation is lower than the mean of domestic inflation, reflecting the process of real appreciation of the Mexican peso during the period.

**Vector Autoregression Analysis of Inflation Sample Statistics, June 1998–March 1991**

(Monthly rates of change)

**Vector Autoregression Analysis of Inflation Sample Statistics, June 1998–March 1991**

(Monthly rates of change)

Variables | Mean | Standard Error | Minimum | Maximum |
---|---|---|---|---|

P | 1.74 | 0.82 | 0.60 | 4.71 |

ii | 1.18 | 1.12 | -1.70 | 2.96 |

w | 2.32 | 3.07 | -3.29 | 7.61 |

z | 1.17 | 2.81 | 0.00 | 8.67 |

m | 2.77 | 4.48 | -6.78 | 15.65 |

m (seasonally adjusted) | 2.91 | 1.41 | -1.05 | 5.19 |

**Vector Autoregression Analysis of Inflation Sample Statistics, June 1998–March 1991**

(Monthly rates of change)

Variables | Mean | Standard Error | Minimum | Maximum |
---|---|---|---|---|

P | 1.74 | 0.82 | 0.60 | 4.71 |

ii | 1.18 | 1.12 | -1.70 | 2.96 |

w | 2.32 | 3.07 | -3.29 | 7.61 |

z | 1.17 | 2.81 | 0.00 | 8.67 |

m | 2.77 | 4.48 | -6.78 | 15.65 |

m (seasonally adjusted) | 2.91 | 1.41 | -1.05 | 5.19 |

The rate of monetary expansion is the variable with the largest fluctuations. It shows the highest standard error, as well as the widest range of variation in the sample. In Chart 18, it is also possible to observe its strong seasonal pattern. In order to isolate the seasonal component, which does not necessarily affect inflation, the series was seasonally adjusted,^{6} which significantly reduced the variability of the series. The adjusted series was used in the estimation of the model.

The rate of growth of wages also shows a much higher variability than the rate of domestic inflation. In addition, the rate of growth of wages exhibits a zigzag pattern, with high values usually followed by low values and vice versa. The rate of domestic inflation shows peaks in December and January, which are likely to be captured by the dummy variables in the regressions.

### Estimation Results

The system of four equations was estimated by the method of seemingly unrelated regressions.^{7} There is a gain in efficiency from using this method instead of ordinary least squares when the set of explanatory variables is not the same for all the equations in the system, and the errors are correlated across equations. As mentioned above, the set of explanatory variables in the equation for imported inflation differs from the one used in the other three equations, and the errors are likely to be correlated across equations.

The results of the estimation are presented in Table 11. Although these coefficients cannot be given a structural interpretation, they are useful for determining to what extent changes in one variable are associated with changes in other variables. The equation for inflation shows significant positive coefficients for the two dummy variables, presumably capturing the effect of adjustments in public sector prices. Changes in minimum wages, on the other hand, have little effect on domestic inflation, but have a significant effect on the rate of increase of nominal wages.

**Vector Autoregression Analysis of Inflation Estimation Results, June 1988–March 1991**

**Vector Autoregression Analysis of Inflation Estimation Results, June 1988–March 1991**

Equations | ||||
---|---|---|---|---|

P | ii | w | m | |

constant | 0.25 | 0.58 | 1.41 | 1.31 |

(0.23) | (0.29) | (1.57) | (0.45) | |

s1 | 1.00 | -2.55 | 0.88 | |

(0.33) | (2.26) | (0.66) | ||

s12 | 1.21 | -1.50 | 1.21 | |

(0.36) | (2.47) | (0.72) | ||

z | 0.03 | 0.39 | -0.04 | |

(0.03) | (0.21) | (0.06) | ||

ii(t-1) | 0.17 | 0.47 | 0.54 | 0.05 |

(0.08) | (0.20) | (0.55) | (0.16) | |

ii(t-2) | 0.03 | 0.02 | -0.49 | 0.42 |

(0.08) | (0.20) | (0.59) | (0.17) | |

p(t-1) | 0.33 | 0.42 | -0.79 | |

(0.13) | (0.89) | (0.26) | ||

p(t-2) | 0.03 | 0.13 | 0.44 | |

(0.12) | (0.84) | (0.24) | ||

w(t-1) | 0.01 | -0.34 | -0.13 | |

(0.03) | (0.21) | (0.06) | ||

w(t-2) | 0.05 | 0.08 | -0.09 | |

(0.03) | (0.22) | (0.06) | ||

m(t-1) | 0.15 | -0.80 | 0.96 | |

(0.08) | (0.54) | (0.16) | ||

m(t-2) | -0.06 | 0.90 | -0.30 | |

(0.08) | (0.55) | (0.16) | ||

R^{2} | 0.81 | 0.19 | 0.34 | 0.74 |

R^{2} | 0.71 | 0.14 | 0.00 | 0.61 |

SSE | 0.44 | 1.04 | 3.06 | 0.88 |

DW | 1.79 | 1.77 | 1.68 | 1.90 |

Q(15) | 17.05 | 8.53 | 10.40 | 5.64 |

Sig. level | 0.32 | 0.90 | 0.79 | 0.99 |

**Vector Autoregression Analysis of Inflation Estimation Results, June 1988–March 1991**

Equations | ||||
---|---|---|---|---|

P | ii | w | m | |

constant | 0.25 | 0.58 | 1.41 | 1.31 |

(0.23) | (0.29) | (1.57) | (0.45) | |

s1 | 1.00 | -2.55 | 0.88 | |

(0.33) | (2.26) | (0.66) | ||

s12 | 1.21 | -1.50 | 1.21 | |

(0.36) | (2.47) | (0.72) | ||

z | 0.03 | 0.39 | -0.04 | |

(0.03) | (0.21) | (0.06) | ||

ii(t-1) | 0.17 | 0.47 | 0.54 | 0.05 |

(0.08) | (0.20) | (0.55) | (0.16) | |

ii(t-2) | 0.03 | 0.02 | -0.49 | 0.42 |

(0.08) | (0.20) | (0.59) | (0.17) | |

p(t-1) | 0.33 | 0.42 | -0.79 | |

(0.13) | (0.89) | (0.26) | ||

p(t-2) | 0.03 | 0.13 | 0.44 | |

(0.12) | (0.84) | (0.24) | ||

w(t-1) | 0.01 | -0.34 | -0.13 | |

(0.03) | (0.21) | (0.06) | ||

w(t-2) | 0.05 | 0.08 | -0.09 | |

(0.03) | (0.22) | (0.06) | ||

m(t-1) | 0.15 | -0.80 | 0.96 | |

(0.08) | (0.54) | (0.16) | ||

m(t-2) | -0.06 | 0.90 | -0.30 | |

(0.08) | (0.55) | (0.16) | ||

R^{2} | 0.81 | 0.19 | 0.34 | 0.74 |

R^{2} | 0.71 | 0.14 | 0.00 | 0.61 |

SSE | 0.44 | 1.04 | 3.06 | 0.88 |

DW | 1.79 | 1.77 | 1.68 | 1.90 |

Q(15) | 17.05 | 8.53 | 10.40 | 5.64 |

Sig. level | 0.32 | 0.90 | 0.79 | 0.99 |

The significance of the lagged values of the various endogenous variables in accounting for the evolution of each endogenous variable can be seen in Table 12, which presents the results of exclusion restriction tests for each of the regressions. The current level of domestic inflation is affected significantly by lagged values of domestic inflation, imported inflation, and money expansion. The effect of lagged values of wage changes, on the other hand, is weaker. These results indicate that the empirical evidence would not support a structural model of inflation that emphasizes past changes in wages as the main source of current inflation.

**Vector Autoregression Analysis of Inflation Exclusion Restrictions**

**Vector Autoregression Analysis of Inflation Exclusion Restrictions**

Variables | Equations | |||
---|---|---|---|---|

P | ii | w | m | |

P | 12.64 | 0.57 | 9.34 | |

(—) | (0.75) | (0.01) | ||

ii | 7.09 | 7.89 | 1.08 | 9.92 |

(0.03) | (0.02) | (0.58) | (0.01) | |

w | 3.18 | 4.29 | 5.01 | |

(0.20) | (0.12) | (0.08) | ||

m | 4.83 | 2.80 | 61.30 | |

(0.09) | (0.25) | (—) |

**Vector Autoregression Analysis of Inflation Exclusion Restrictions**

Variables | Equations | |||
---|---|---|---|---|

P | ii | w | m | |

P | 12.64 | 0.57 | 9.34 | |

(—) | (0.75) | (0.01) | ||

ii | 7.09 | 7.89 | 1.08 | 9.92 |

(0.03) | (0.02) | (0.58) | (0.01) | |

w | 3.18 | 4.29 | 5.01 | |

(0.20) | (0.12) | (0.08) | ||

m | 4.83 | 2.80 | 61.30 | |

(0.09) | (0.25) | (—) |

Lagged levels of imported inflation have a substantial effect on the current level of imported inflation. The current rate of wage increase is affected significantly only by lagged wage increases. This implies that a model that stresses the role of past inflation as the main determinant of current wage changes, owing for example to lags in adjusting real wages or to expectations about future inflation, would be inconsistent with the data. The current level of monetary expansion, on the other hand, is affected significantly by lagged values of all the endogenous variables. This is consistent with the presumption that monetary expansion should be considered an endogenous variable.

The ability of the model to account for the behavior of the endogenous variables during the estimation period can be assessed from the goodness-of-fit measures presented in Table 11, and from Chart 19. Estimated values for domestic inflation and monetary expansion follow actual values closely. This, however, is not the case for imported inflation and for wage changes.

In the case of imported inflation, the poor fit is not surprising, because it is mainly due to the difficulty found in modeling monthly changes in exchange rates between countries with floating or managed exchange rates. Imported inflation in Mexico is composed of the nominal effective depreciation of the Mexican peso with respect to the currency of a group of countries, plus the weighted average inflation of those countries. Since inflation in those countries behaves smoothly, most of the variation in imported inflation is due to changes in the nominal effective exchange rate. Given the crawling peg policy for the Mexican peso with respect to the U.S. dollar, sharp changes in the nominal effective exchange rate reflect exchange rate changes between the U.S. dollar and the other currencies in the basket, which are difficult to model.

The poor fit of the equation for wage changes is surprising, however. Both past inflation and past wage changes were expected to be important influences on the behavior of current wage changes. Past inflation would be important if there were some formal or informal backward-looking wage-indexation mechanism. Even in the presence of forward-looking indexation, past inflation would be relevant if it helped to predict future inflation. Past wage changes would be important if there were some inertia in wage determination. However, in the case of inertia lagged wage changes would have a positive effect on current wage changes, in contrast to the one-lag negative effect in the estimation results.

These unexpected results regarding the equation for wage changes may reflect the way in which the information on wages is collected, and the coverage of this series. Wage data are obtained monthly from a group of firms. In a given month, however, only about half the firms in the group are sampled, and the rest is sampled the following month. This alternation of sample sets may introduce some bias that produces the zigzag pattern shown by wage changes. Furthermore, the series on wages also includes some benefits that are paid during the year. To the extent that those benefits are not distributed uniformly throughout the year, this may introduce an additional bias in the estimation.

### Dynamic Response to Innovations

VAR models may be used to derive the dynamic response of the endogenous variables to the various innovations. As indicated above, the first step is to recover the innovations from the residuals of the estimation of the reduced form equations, under a suitable set of additional assumptions. Explaining this procedure, however, requires that we refer to some of the concepts presented in the Appendix.

In terms of equations (1) and (2) in the Appendix, we want to derive the dynamic response of *y*(*t*) to the innovations *e*(*t*), having estimated (1). To do this, it is necessary to model the relationship between the errors in equation (1) and the innovations in equation (2). In other words, it is necessary to estimate (*I-B*_{0}) in equation (5). The elements in the matrix *(I-B*_{0}) indicate to what extent the various shocks have an immediate effect on the various endogenous variables.

The usual way of modeling the relationship between the reduced form errors *u*(*t*) and the innovations *e*(*t*) has been to use a Cholesky factorization, which assumes that (*I-B*_{0}) is a lower triangular matrix, thereby implying a recursive structure for the immediate effect of the various innovations.^{8} Thus, the first variable is affected immediately only by the first innovation, the second variable by the first and the second innovation, and so on. However, Bernanke has suggested that the assumption of a recursive structure is rather restrictive and proposed a method for estimating more general relationships between *u*(*t*) and *e*(*t*).^{9} This method can be used either with an exactly identified system, or with an overidentified system and then tested for the overidentification restrictions.^{10}

An overidentified system was estimated for the residuals from the model above. In this system, domestic inflation was assumed to be affected immediately by all the innovations, while wage changes and monetary expansion were assumed to be affected within the same month only by their own innovations.^{11} The results from the estimation are

where *u*_{ii} = *e*_{ii}, *u*_{w} = *e*_{w}, and *u*_{m}, denote the reduced form residual and the structural form innovation, respectively, for variable *j*, and the numbers in parentheses indicate standard errors. The coefficients for innovations in imported inflation and monetary expansion have the expected positive sign, but they are not statistically significant. The coefficient on wage innovations, on the other hand, is significant, but it has a negative sign, which is difficult to account for. It implies that a positive innovation in wages causes domestic inflation in the same month to be lower than otherwise. This may be just another consequence of the peculiar behavior shown by the series on wages, as mentioned before.^{12} A chi-square test indicates that the overidentification restrictions cannot be rejected at the usual levels of significance. With three degrees of freedom the chi-square statistic was 1.35, implying a significance level of 0.72.^{13}

The dynamic effect of innovations in domestic inflation, imported inflation, wage increases, and monetary expansion are presented in Charts 20-23. In each case, the graphs show the dynamic response over a year to an initial shock of the size of one standard deviation of the innovation under consideration.^{14}

**Impulse Response to Innovation in Domestic Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Domestic Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Domestic Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Imported Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Imported Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Imported Inflation**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Wages**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Wages**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Wages**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Money**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Money**

(Monthly rates of change, in percent)

**Impulse Response to Innovation in Money**

(Monthly rates of change, in percent)

The magnitude of the innovation in domestic inflation is relatively small when compared with the other innovations. Its effect on domestic inflation declines quickly, practically disappearing after two months. It has a negative effect on monetary expansion for about five months, and no effect on imported inflation, because imported inflation is exogenous. It has a positive effect on wage increases in the two months following the shock, and a smaller negative effect afterward.

The innovation in imported inflation has a larger and more persistent effect on all the variables. Domestic inflation increases for about nine months, while monetary expansion increases for about seven months and then declines somewhat. It also has a persistent effect on wage increases, with a zigzag pattern for the first few months following the shock.

The magnitude of the innovation in wages is the largest one, but its effect on domestic inflation is relatively small. It has no effect on imported inflation, and a negative effect on money growth that lasts for about six months and is followed by a smaller positive effect. It produces a marked zigzag pattern for wage increases that lasts for several months after the shock. Given the relative small effect on domestic inflation, changes in real wages follow changes in nominal wages.

The innovation in monetary expansion has a relatively small effect on domestic inflation that lasts for about five months, and no effect on foreign inflation. It has a persistent effect on wage increases, with a zigzag pattern during the first few months following the shock. Monetary expansion increases for about four months, and then declines somewhat for the next six months.

### Decomposition of Variance of Forecast Error

One way of assessing the relative importance of each innovation in causing movements in a given endogenous variable is to calculate the proportion of the variance of the forecast error for that variable that can be attributed to each of the innovations. Innovations with large fluctuations and with large effects on the endogenous variable will account for a large proportion of the variance of the forecast error. These calculations are presented in Table 13 for several forecast horizons.

**Vector Autoregression Analysis of Inflation: Decomposition of Variance**

**Vector Autoregression Analysis of Inflation: Decomposition of Variance**

Series | Horizon (months) | Standard Error | Innovations | |||
---|---|---|---|---|---|---|

e_{p} | e_{ii} | e_{w} | e_{m} | |||

Domestic inflation | 1 | 0.35 | 90 | 1 | 9 | 0 |

3 | 0.47 | 56 | 27 | 8 | 9 | |

6 | 0.53 | 45 | 40 | 7 | 8 | |

9 | 0.53 | 44 | 41 | 7 | 8 | |

12 | 0.53 | 44 | 41 | 7 | 8 | |

Imported inflation | 1 | 0.99 | 0 | 100 | 0 | 0 |

3 | 1.12 | 0 | 100 | 0 | 0 | |

6 | 1.13 | 0 | 100 | 0 | 0 | |

9 | 1.13 | 0 | 100 | 0 | 0 | |

12 | 1.13 | 0 | 100 | 0 | 0 | |

Wages | 1 | 2.46 | 0 | 0 | 100 | 0 |

3 | 2.87 | 1 | 5 | 89 | 5 | |

6 | 2.90 | 1 | 5 | 88 | 6 | |

9 | 2.91 | 1 | 6 | 87 | 6 | |

12 | 2.92 | 1 | 6 | 87 | 6 | |

Money | 1 | 0.70 | 0 | 0 | 0 | 100 |

3 | 1.23 | 8 | 5 | 13 | 74 | |

6 | 1.48 | 6 | 22 | 18 | 54 | |

9 | 1.50 | 6 | 23 | 18 | 53 | |

12 | 1.51 | 6 | 23 | 18 | 53 |

**Vector Autoregression Analysis of Inflation: Decomposition of Variance**

Series | Horizon (months) | Standard Error | Innovations | |||
---|---|---|---|---|---|---|

e_{p} | e_{ii} | e_{w} | e_{m} | |||

Domestic inflation | 1 | 0.35 | 90 | 1 | 9 | 0 |

3 | 0.47 | 56 | 27 | 8 | 9 | |

6 | 0.53 | 45 | 40 | 7 | 8 | |

9 | 0.53 | 44 | 41 | 7 | 8 | |

12 | 0.53 | 44 | 41 | 7 | 8 | |

Imported inflation | 1 | 0.99 | 0 | 100 | 0 | 0 |

3 | 1.12 | 0 | 100 | 0 | 0 | |

6 | 1.13 | 0 | 100 | 0 | 0 | |

9 | 1.13 | 0 | 100 | 0 | 0 | |

12 | 1.13 | 0 | 100 | 0 | 0 | |

Wages | 1 | 2.46 | 0 | 0 | 100 | 0 |

3 | 2.87 | 1 | 5 | 89 | 5 | |

6 | 2.90 | 1 | 5 | 88 | 6 | |

9 | 2.91 | 1 | 6 | 87 | 6 | |

12 | 2.92 | 1 | 6 | 87 | 6 | |

Money | 1 | 0.70 | 0 | 0 | 0 | 100 |

3 | 1.23 | 8 | 5 | 13 | 74 | |

6 | 1.48 | 6 | 22 | 18 | 54 | |

9 | 1.50 | 6 | 23 | 18 | 53 | |

12 | 1.51 | 6 | 23 | 18 | 53 |

For all the variables, the forecast standard error tends to some upper bound as the forecast horizon lengthens. From the magnitude of the standard errors, it is clear that this model would generate more precise forecasts for domestic inflation, and less precise forecasts for wage changes, than for the other variables in the system.

The variance of the forecast error for domestic inflation is almost entirely explained by innovations in domestic inflation and in imported inflation. Innovations in wages and in monetary expansion combined account for only about 15 percent of total variance. While innovations in domestic inflation are the most important factor for all horizons, innovations in imported inflation are almost as important for six-month and longer horizons.

The only important component in the decomposition of the forecast error for wage increases is wage innovations. All the other innovations taken together account for only 13 percent of the variance of the forecast error. This is consistent with the results of the exclusion restriction tests presented in Table 12, where only past wage increases have a significant effect on current wage changes, and with our assumption that innovations in other variables do not have an immediate effect on wage changes.

The variance of the forecast error for monetary expansion is explained by its own innovation, and by innovations in imported inflation and in wages, with virtually no contribution from innovations in domestic inflation. While monetary expansion innovations dominate for short horizons, imported inflation and wage innovations taken together also become important for six-month and longer horizons.

### Decomposition of the Forecast Error for 1990–91

The model can also be used to assess the contribution of each of the innovations in producing the actual fore-cast error for a given period within the estimation sample. Charts 24 and 25 present such decomposition for the period January 1990-March 1991. Chart 24 shows the actual inflation rate during that period, and the inflation rate that would have been forecast using the estimated model and information on p, ii, w, and *m* up to December 1989.^{15} The vertical difference between the two lines represents the forecast error. This error is attributed to the various innovations in Chart 25, where the vertical sum of the various lines adds up to the forecast error.

**Historical Decomposition of Forecast Error, January 1990-March 1991**

(Monthly rates of change, in percent)

**Historical Decomposition of Forecast Error, January 1990-March 1991**

(Monthly rates of change, in percent)

**Historical Decomposition of Forecast Error, January 1990-March 1991**

(Monthly rates of change, in percent)

From Chart 24, it is clear that actual inflation was higher than forecasted inflation for most of this period. The forecast error is positive and particularly large for June, July, and November 1990, and negative and significant for January 1991. According to Chart 25, the June 1990 error was mostly due to wage innovations, while the July 1990 error was due to domestic inflation innovations. The November 1990 error was due to both imported inflation and domestic inflation innovations. Imported inflation was higher than forecasted for most of the period, which is reflected in its positive contribution to the fore-cast error. The positive contribution of domestic inflation innovations in November 1990, and their negative contribution in January 1991, may be due to a modification in the calendar for announcing new measures under the PECE with respect to past experience. While in previous years adjustments in public sector prices and wages were announced in December, in 1990 these adjustments were announced in November. Therefore, the main impact of the new measures may have taken place in November and December 1990, rather than in December 1990 and January 1991, as predicted by the dummies used in the forecasting exercise.

As a final exercise, the model was used to forecast beyond the estimation period. With the model estimated with information up to March 1991, and with the values of the endogenous variables observed up to that month, the forecasted cumulative inflation for the next eight months was 9.0 percent, while the actual cumulative inflation was 9.3 percent.^{16}

## Conclusions

During the period June 1988-March 1991, the monthly rate of inflation in Mexico responded significantly to adjustments in public sector prices and to lagged values of domestic inflation, imported inflation, and monetary expansion, but showed little reaction to lagged values of wage changes. Regarding contemporaneous effects, the response to imported inflation and monetary expansion was weak, while the reaction to wage changes was significant but negative, which is difficult to explain.

The evolution of nominal wages during this period responded primarily to changes in the minimum wage rate, and to lagged changes in wages, but showed relatively small reaction to lagged values of domestic inflation, imported inflation, and monetary expansion. Monetary expansion, on the other hand, responded to lagged values of all the other variables.

The dynamic effects of the various innovations were derived under the assumption that domestic inflation adjusts within the same month to all types of innovations, while wage increases and monetary expansion adjust with a one-month lag to innovations in other variables. The results indicate that innovations in imported inflation have a significant and persistent effect on domestic inflation, while innovations in domestic inflation have a significant but less persistent effect. The effects of innovations in monetary expansion and wage changes are considerably smaller.

These results have a number of implications regarding the appropriate modeling of transmission channels in a structural model for inflation. Inflation should not be modeled as being determined primarily by a markup over wages, since inflation is associated with other variables, and the contemporaneous effect of wage changes seems to be negative rather than positive. In addition, modeling wage adjustments as responding predominantly to past changes in prices would fail to account for a large fraction of the variability in wage changes. Also, monetary expansion should not be considered an exogenous variable, since it responds to the other endogenous variables in the system.

Clearly, these results depend on the particular identifying restrictions regarding the contemporaneous effects of the various innovations, and on the particular set of variables included in the model. While modifying the identifying restrictions would not affect significantly many of the results, modifying the set of variables might do so.

Regardless of the particular model used, attempts to identify the various forces that determine inflation in Mexico on a monthly basis must, however, confront two problems. One is the peculiar behavior of the data on wages, for which a satisfactory explanation is not readily available. There is the possibility that the quality of the data is poor, and thus the conclusions could be based on erroneous data. The other problem is that many exogenous shocks in Mexico take place simultaneously. Usually, minimum wages and public sector prices are adjusted, and a new path for the exchange rate is announced at the end of the year. This problem of simultaneous shocks at the end of the year is compounded by the adjustments that must be made to the series on monetary expansion, because of seasonally, and to the series on wage changes, because of the December bonus. To the extent that these adjustments are not perfect, they may introduce additional biases in the estimation.

## Appendix I Description of Model

Denoting the endogenous variables by *y*_{1} …, *y*_{n} and the exogenous variables by *x*_{1} …, *x*_{m}, the system to estimate is

where *y*(*t*) is an *nxl* vector of endogenous variables at time *t, x*(*t*) is an *mxl* vector of exogenous variables at time *t, C* is an *nxm* matrix of coefficients associated with the exogenous variables’ lagged j periods, *Dj* is an *nxn* matrix of coefficients associated with the endogenous variables, lagged j periods, and *u*(*t*) an *nxl* vector rep-resenting the reduced form errors.

The system described by equation (1) can be interpreted as resulting from a structural model such as

where *A*_{j} is an *nxm* matrix, *B*_{0} is an *nxn* matrix with diagonal elements equal to zero, *B*_{j} (for *j* > 0) is an *nxn* matrix, and *e* (*t*) is an *nxl* vector of structural innovations, which are assumed to have zero cross-correlation. The relationship between the reduced form system in equation (1) and the structural system in equation (2) is given by

where *I* is the nxn identity matrix.

According to equation (2), the dynamics of the endogenous variables are driven by the realization of the innovations *e*(*t*), and the evolution of the exogenous variables *x*(*t*). The innovations *e*(*t*) represent exogenous shocks to the variables *y*(*t*), which are not captured by the exogenous variables *x*(*t*).

From the estimated reduced form errors (the estimated *u*(*t*)), it is possible to recover the innovations (the *e*(*t*)) by estimating (*I*-*B*_{0}) in equation (5) under certain additional assumptions. These assumptions refer to the contemporaneous response of the endogenous variables to the various innovations. To the extent that the ith variable does not respond to the jth innovation within the same period, the (i, *j*) element of the (*I*-*B*_{0}) matrix is equal to zero. The structural errors can be recovered from the reduced form errors by choosing a sufficient number of off-diagonal elements in the (*I*-*B*_{o}) matrix to be zero.

## Bibliography

Bernanke Ben S., “Alternative Explanations of the Money-Income Correlation,”

, Vol. 25 (1986), pp. 49–100.*Carnegie-Rochester Conference Series on Public Policy*Cooley Thomas F., and Stephen F., LeRoy “A Theoretical Macroeconometrics: A Critique,”

, Vol. 16 (1985), pp. 283–308.*Journal of Monetary Economics*Guerrero Victor M., and Luis G. Arias “Analisis de la Inflacion en Mexico de 1970 a 1987 Mediante Vectores Autorregresivos,”

, Vol. 57 (1990), pp. 379–401.*El Trimestre Economico*Ize Alain, and Gabriel Vera eds.,

(Mexico: El Colegio de Mexico, 1984).*La Inflacion en Mexico*Kaminsky Graciela, “Dual Exchange Rates: The Mexican Experience 1982–1987,” a paper presented at an IMF Seminar on April 2, 1992.

Learner Edward E., “Vector Autoregression for Causal Inference?,”

, Vol. 22 (1985), pp. 255–304.*Carnegie-Rochester Conference Series on Public Policy*Leiderman Leonardo, “On the Monetary-Macro Dynamics of Colombia and Mexico,”

, Vol. 14 (1984), pp. 183–201.*Journal of Development Economics*Lütkepohl Helmut, “Comparison of Criteria for Estimating the Order of Vector Autoregressive Process,”

, Vol. 6 (1985), pp. 35–52.*Journal of Time Series Analysis*Puente Francisco, “Resultados Preliminares del Uso de un Modelo “VAR” para Analizar el Proceso de Inflacion en Mexico,”

(Banco de Mexico), Vol. 15 (1989), pp. 23–35.*Boletin de Economia Internacional*Sims Christopher A., “Are Forecasting Models Usable for Policy Analysis?”

, Vol. 10 (Winter 1986), pp. 2–16.*Quarterly Review*, Federal Reserve Bank of Minneapolis

^{}1

For a discussion about divergent views regarding the valid uses of vector autoregressions, see Cooley and LeRoy (1985), Learner (1985), and Sims (1986). Vector autoregressions have been used to analyze Mexican inflation for periods prior to the one examined in this section. See, for example, the papers included in Ize and Vera (1984), Leiderman (1984), Puente (1989), Guerrero and Arias (1990), and Kaminsky (1991).

^{}2

Although it could be argued that policy variables were really endogenous because they were adjusted according to the evolution of other variables in the economy, those adjustments were very infrequent in terms of monthly observations. Modeling the timing and size of those adjustments as endogenous would require a substantial effort in relation to the benefits to be obtained, and is not attempted here.

^{}4

M2 is defined as private sector holdings of currency, demand deposits, and short-term banking instruments. It is not clear which is the most appropriate monetary aggregate to include in the estimation. M2 was preferred over Ml because Ml showed some large increases during part of 1990, owing to changes in regulations that are not necessarily associated with changes in the rate of inflation or other variables included in the model.

^{}5

The countries included in the group and their respective weights are United States (0.638), Japan (0.109), Germany (0.079), France (0.038), Canada (0.037), United Kingdom (0.035), Italy (0.033), and Spain (0.030).

^{}6

The series was seasonally adjusted by the exponential smoothing technique included in the RATS computer package, using a model with an additive seasonal term and no trend.

^{}7

In order to test for stationarity of the data, an augmented Dickey-Fuller test was applied to each series. Nonstationarity could be rejected at the 5 percent level of significance for the series on domestic inflation, and wage increases. For the series on money growth and foreign inflation, nonstationarity could only be rejected at the 25 percent and 30 percent levels of significance, respectively. Although these results indicate that the rejection of nonstationarity is not definite for money growth and foreign inflation, these series were not differenced before estimation due to strong priors about the relationship between these variables and domestic inflation. In order to determine the number of lags to include in the estimation, four different criteria were applied to the data for a range from zero to six lags. Two criteria, Hannan-Quinn and Schwarz, indicated that the appropriate lag length was one. The FPE criterion indicated a lag length of two, while the Akaike criterion indicated a lag length of six, thereby using up almost all the degrees of freedom. Based on these results, two lags were included in the estimation. For a comparison of the various criteria see Lütkepohl (1985).

^{}8

This method is applied, for example, by Leiderman (1984), Puente (1989), and Guerrero and Arias (1990).

^{}9

See Bernanke (1986). This method is applied, for example, by Kaminsky (1992).

^{}10

Since there are four types of innovations, the covariance matrix of the reduced-form contains ten independent moments. Since identification is achieved here by assuming that some innovations have no contemporaneous effect on some endogenous variables, whether the model is exactly identified or overidentified depends on whether six or more off-diagonal elements of the *(I-B*_{0}) matrix are assumed to be equal to zero. If the model is overidentified, the decomposition will not be able to replicate exactly the correlation matrix of the residuals. The test for overidentification checks to what extent this departure is significant.

^{}12

A possible explanation for this result could be based on productivity shocks. A positive productivity shock, for example, would lead to an increase in wages, while simultaneously increasing output and thus reducing inflationary pressures. This explanation, however, would require a rather curious process generating productivity shocks to account for the magnitude and behavior of observed monthly wage changes.

^{}13

Alternative systems were estimated for the residuals, where monetary expansion was also assumed to be affected immediately by other innovations, but the results were unsatisfactory. For an exactly identified system, the likelihood function was ill-behaved. For overidentified systems, the overidentification restrictions were rejected at the 10 percent level of significance, with one exception. The restriction was not rejected when only imported inflation innovations were assumed to affect monetary expansion in the same period. However, in this case the estimated coefficient is small and statistically nonsignificant, so that the results are essentially the same as those presented in this section.

^{}14

All the graphs, except one, are drawn using the same vertical scale so as to facilitate a visual comparison of the size of the various shocks and the magnitude of the response of the various variables. The one exception is the graph on the behavior of wages upon an innovation in wages, in Chart 22. In this case, the vertical scale was reduced by half because changes in wages show very large fluctuations when compared with other cases and other variables.

## OCCASIONAL PAPERS: Recent Occasional Papers of the International Monetary Fund

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90. The Internationalization of Currencies: An Appraisal of the Japanese Yen, by George S. Tavlas and Yuzuru Ozeki. 1992.

89. The Romanian Economic Reform Program, by Dimitri G. Demekas and Mohsin S, Khan. 1991.

88. Value-Added Tax: Administrative and Policy Issues, edited by Alan A. Tait. 1991.

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84. Financial Liberalization, Money Demand, and Monetary Policy in Asian Countries, by Wanda Tseng and Robert Corker. 1991.

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77. Determinants and Systemic Consequences of International Capital Flows, by Morris Goldstein, Donald J. Mathieson, David Folkerts-Landau, Timothy Lane, J. Saul Lizondo, and Liliana Rojas-Suarez. 1991.

76. China: Economic Reform and Macroeconomic Management, by Mario Blejer, David Burton, Steven Dunaway, and Gyorgy Szapary. 1991.

75. German Unification: Economic Issues, edited by Leslie Lipschitz and Donogh McDonald. 1990.

74. The Impact of the European Community’s Internal Market on the EFTA, by Richard K. Abrams, Peter K. Cornelius, Per L. Hedfors, and Gunnar Tersman. 1990.

73. The European Monetary System: Developments and Perspectives, by Horst Ungerer, Jouko J. Hauvonen, Augusto Lopez-Claros, and Thomas Mayer. 1990.

72. The Czech and Slovak Federal Republic: An Economy in Transition, by Jim Prust and an IMF Staff Team. 1990.

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