Much of the existing work has been very limited in coverage of commodities and somewhat limited in coverage of locations (ie, focusing on Europe). We use the result of the Monte Carlo procedure to adjust our baseline estimates for each individual commodity. Thus, the typical product half-life is still below the consensus limit in the PPP literature.
This finding suggests that at least part of The answer to the PKM puzzle lies in the difference between the price dynamics at the level of individual goods and the price dynamics of the aggregate price index. Normalization to the average world price avoids the arbitrary choice of numeraire location.3 Qitjs are called 'deviations according to the law of one price': when multiplied by 100, they represent the percentage by which the price of an individual commodity at a given location on a given date differs from the average world prices of the same commodity on the same date. If the law of one price held exactly for all goods and locations at date t = τ, the entire mass distribution would be at 0.
We are effectively calculating a rough estimate of the purchasing power parities of the cities in our sample. One surprising feature of the data is the fact that the spread of non-traded goods prices is lower across the US.
Dynamics of Law-of-One-Price deviations
Our main interest is ρ which represents the speed of price adjustment or the persistence of price deviations. The international estimates change somewhat in the distribution, but not much in terms of the mean. For both the international and intranational data, the hypothesis is rejected for only a small portion of the goods in our sample.
The persistence of deviations from the mean in the international data increases from 0.51 to 0.91, which translates into a half-life of more than 7 years. Deviations from the Law of One Price remain somewhat less persistent in the least developed countries than in the OECD. We firmly reject the absolute convergence hypothesis in the international context: for 86% of the goods we reject it at the 1% level of significance.
The role of the constant term in the PKM literature is different, reflecting the fact that when index numerical data are used, the degree of absolute deviation from the PKM is unknown. The aim of the next section is to reconcile these successive results and, in a sense, to re-establish the connection between purchasing power parity and its basic building block – the law of one price.
Reconciliation with macroeconomic evidence
We allow the key parameters of the data generating process to vary depending on the individual good, and use the corresponding experiment outcome to perform a bias correction on the baseline persistence estimate of each individual good. First, note that for a given T, the magnitude of the bias in the GMM estimator depends on: The basic axes are the true persistence and variance ratio of the data generating process and the vertical axis is the estimated bias.
It also includes the summary of the goodness-of-fit estimates of the variance ratio σ2η/σ2v.7 The bottom panel contains the adjusted bias estimates. We see, first of all, that small sample bias is substantial in all cases, but none of the relative robustness rankings are affected by the bias correction. See Arellano (2003) for the estimator formula of σ2η and σ2v. studies using the real exchange rate based on the CPI.
The first type of aggregation bias arises when the aggregate data is a simple, equally weighted average of the underlying time series. Since more durable goods have more volatility in the unconditional sense, it follows that the stability of the real aggregate exchange rate will be biased in the direction of these goods. A simple calculation reveals that the first-order autocorrelation of the aggregate of the individual deviations of the Law of One Price, qit = 0.5qHit + 0.5qitL, becomes.
Based on the above formula, the weight of each of the high (low) persistence commodities in the aggregate level of persistence would be rather implied by their sample proportions.8 Aggregate persistence would be 0.82, much higher than the simple average stability of the individual deviation levels of the Law of a Price, which is 0.59. The first is what we had before, the cost-weighted average of the fundamental deviations of the Law of One Price. The second term is proportional to the relative price of two goods within a country.
Note that the BLS are more appropriate weights because they apply to cities and therefore correspond to the locations of the price surveys. The first factor is pure aggregation bias in the sense that an equally weighted average of underlying series will generally have different persistence properties than the average persistence of the series being aggregated. The top panel of Table 7 contains all aggregation results for the OECD, LDCs and the US.
For the baseline estimates, we see that the average persistence level across goods is indeed a poor predictor of the persistence of PPP deviations in all cases except the LDC. The gist of the pooled results, after correcting for small sample bias, is that we largely reproduced the persistence levels in the existing literature using our CPI constructs from micro data.
Conclusions
Fourth, our advice to quantitative theorists is to take advantage of the information in micro-data to calibrate and evaluate micro-based models. We also hope to add international data as the limited number of locations combined with the short time sample limited our ability to make accurate inferences for the US. The table below reports the countries included, the dates of the name change, the terms of the change, and the number of observations included.
The total number of observations affected by these reforms that also met our criteria was 3,550, accounting for 64.8% of the 5,480 price changes observed. The next step was trickier, where each of the remaining 1,930 price observations was explicitly examined for evidence of error. After aggregating this subset, we took one of the following actions: i) if it is clear that units have changed, we adjust to common units; ii) if it is clear that the first digit is missing, we enter the digit that is the same as the first digit of the previous and subsequent series; and iii) if it's not clear whether it's a unit change or a missing digit, we do one of two things.
Of the 305 adjustments, we change the units or add the missing figures in 179 cases, and in 126 cases of clear outliers (more than 100% change compared to the total intake from period t−1 to t with return to a level close to that at date t−1) we replaced the period t observation with the average price over the periods t−1 and t+ 1. Lai, 2000, “On cross-country differences in the persistence of real exchange rates”, Journal of Internationale Economy. Mark and Donggyu Sul, 2003, “The dominance of downward bias in estimates of the half-life of PPP anomalies”, stencil, University of New Hampshire.
14] Engel, Charles, 1993, “Real Exchange Rates and Relative Prices: An Empirical Investigation”, Journal of Monetary Economics 32, 35-50. Rose, 1996, "A Panel Project on Purchasing Power Parity: Average Returns Within and Across Countries," Journal of International Economics. 30] Parsley, David and Shang-Jin Wei, 1996, "Convergence to the Law of One Price Without Trade Barriers or Currency Fluctuations", Quarterly Journal of Economics.
Harare, Zimbabwe (200)† Caracas, Venezuela (238)† Washington DC (255) Note: Entries are the city where price data are collected, the country to which the city belongs, and the number of commodities in the analysis for which that city is used. The number of tests/goods is the total number of unit root test statistics calculated for each good, while the median number of sites refers to the number of cross-sectional samples used to calculate each test statistic. The mean shows the averages of the good specific estimates of ρ. means and standard errors are in parentheses.
The number of items is the total number of item-specific estimates of ρ, while the median of the number of cities refers to the number of cross-sectional samples used in the estimation of each item speciÞcρ. Notes: See notes to Table 4. The bottom panel contains the proportion of items for which the null hypothesis of no individual effects is rejected using Hausman tests constructed from GMM estimates under conditional and absolute convergence.
