Lecture 4 Statistical Approach in Index Numbers

Transcription

1 Lecture 4 Statistical Approach in Index Numbers Economics 5415 Index Number Theory Kam Yu Winter 2017

2 Outline 1 Introduction Theil s Probabilistic Approach 2 Basic Concept of Core Inflation 3 Measuring Core Inflation CPIX Trimmed Mean Variance-Weighted Price Index Dynamic Factor Index 4 Evaluations and Discussions Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

3 Required Readings Theil, Henri (1967) Economics and Information Theory, Chicago: Rand McNally and Company, pp Wynne, Mark A. (2008) Core Inflation: A Review of Some Conceptual Issues, Federal Reserve Bank of St. Louis Review, May/June, Part 2, Khan, Mikael, Louis Morel, and Patrick Sabourin (2015) A Comprehensive Evaluation of Measures of Core Inflation for Canada, Bank of Canada Discussion Paper Many of the references can be found on the reading list. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

4 Introduction The Statistical Approach In the test approach we look at mathematical properties of index number formulae. In the economic approach we concentrate on consumer behaviours and their implications for index numbers. Now we examine index numbers from a probabilistic and statistical perspective. This approach is of interests to central bankers, whose mandates are to use monetary policy to stabilize inflation, output, or unemployment. The idea of the so-called core inflation can be different from the principle of a cost of living index. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

5 Introduction Theil s Probabilistic Approach Henri Theil s Probabilistic Approach Suppose we want to compare the average price levels in two cities, say Vancouver (city 1) and Toronto (city 2). The observed prices and quantities of N homogeneous goods and services in Vancouver are p 1 = (p 1 1, p 1 2,..., p 1 N ), x 1 = (x 1 1, x 1 2,..., x 1 N ), and similar for those of Toronto with superscript 2. For any good i, the rate of change of the price if we move from Vancouver to Toronto can be approximated by the logarithmic difference of their prices. That is, π i = log p 2 i log p 1 i = log p2 i pi 1. In the intertemporal case, 1 and 2 are two time periods in the same city. Then the above log difference is the inflation rate of good i. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

6 Introduction Theil s Probabilistic Approach From Vancouver to Toronto Suppose that we pick a good or service randomly in Vancouver. The probability of a dollar spent on good i is equal to its expenditure share s 1 i. The overall change in price level from Vancouver to Toronto, from the perspective of a consumer in Vancouver, is then Π 1 = N i=1 si 1 log p2 i pi 1. The price index from a Vancouver consumer s perspective is obtained by taking the antilogarithm of the above expression: P 1 (p 1, p 2, x 1, x 2 ) = e Π1 = N ( p 2 i p 1 i=1 i ) s 1 i. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

7 Introduction Theil s Probabilistic Approach From Thunder Bay to... But of course the above analysis is one-sided. Let s suppose that a resident in Thunder Bay is considering relocating to either Vancouver or Toronto. Her probability of moving to Vancouver is λ, while the probability of moving to Toronto is 1 λ. The probability of spending a dollar on good i in either city is therefore s i = λs 1 i + (1 λ)s 2 i, where si 2 is the expenditure share of good i for a typical consumer in Toronto. With no a priori preferences, she takes λ = 1/2, which implies that s i = 1 2 (s1 i + s 2 i ). Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

8 Introduction Theil s Probabilistic Approach And the Price Index is... From the perspective of an outsider, the average of log price ratios of the two cities is N Π = s i log p2 i pi 1. i=1 The spatial price index from Vancouver to Toronto is P T (p 1, p 2, x 1, x 2 ) = e Π = N ( p 2 i p 1 i=1 i ) 1 2 (s1 i +s2 i ), which is the Törnqvist price index. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

9 Introduction Theil s Probabilistic Approach Worldwide Cost of Living 2016 The ten most expensive cities in the world Country City WCOL index (New York=100) S ities ove u the ran ing even if lo al ri es re ain sta le Source: The Economist Intelligence Unit Limited An increase in the cost of living in many US locations has seen two US cities move into the top ten most expensive cities in the world, with Western European locations still making up one-half of the total. Three Asian cities complete the top ten. But even in this relationship, the dynamics have changed over Rank Rank movement Singapore Singapore Switzerland Zurich Hong Kong Hong Kong Switzerland Geneva France Paris UK London US New York Denmark Copenhagen South Korea Seoul US Los Angeles Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

10 Introduction Theil s Probabilistic Approach Statistical Quantity Index The above argument can be applied to the quantity index between two cities, which implies the use of the Törnqvist quantity index Q T (p 1, p 2, x 1, x 2 ) = N ( x 2 i x 1 i=1 i ) 1 2 (s1 i +s2 i ). Recall that P T and Q T do not satisfy the product identity. Therefore we choose one of the conjugates (P T, Q T ) or ( P T, Q T ). Therefore we justify the use of the the Törnqvist indices or the geometric indices depending on the assumption on the probability distribution of the price ratios. For more detailed discussions, consult Selvanathan and Prasada Rao (1994) and Diewert (2010). Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

11 Basic Concept of Core Inflation What Do Central Bankers Want? The general inflation rate is an important statistic that central bankers rely on to conduct monetary policy. In countries like Australia, Canada, Finland, the U.K., etc, the objective of the central bank is inflation targeting. The effects of any monetary policy, however, have time lags on inflation and output after the policy changes. A central bank therefore wants to measure the underlying trend in price movement, not the monthly fluctuations in the monthly CPI. The so-called core inflation, however, is not always clearly defined. Different writers have different interpretation, depending on the assumptions in their macroeconomic models. There are a variety of methods to capture the underlying inflation in the short run for central bankers. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

12 Basic Concept of Core Inflation A VAR Macroeconomic Analysis Quah and Vahey (1995) define core inflation as the component of measured inflation that has no medium to long run impact on real output. This view is consistent with the theory that nominal shocks to the economy have no impact on the real variables. In this case it reflects a vertical long run Phillips curve. In particular, there are two types of exogenous shock to measured inflation. The first type, η 1, has no impact on output after some fixed periods. The second type, η 2 have medium to long run effects on output. Let log Q(t) and log P(t) be the output growth rate and inflation rate from period t 1 to period t, where Q(t) and P(t) are the quantity and price index of the aggregate economy. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

13 Basic Concept of Core Inflation Output and Inflation Dynamics Define the two-dimensional vectors [ ] log Q(t) x(t) =, η(t) = log P(t) The dynamic process is [ ] η1 (t). η 2 (t) x(t) = D(0)η(t) + D(1)η(t 1) + (1) = D(j)η(t j), j=0 where D(j) is a two-dimensional square matrix, and the variance-covariance matrix of η(t) is assumed to be Var(η(t)) = I. That is, η 1 and η 2 are uncorrelated for all leads and lags, and their variances are normalized to 1. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

14 Basic Concept of Core Inflation Core Inflation Since we assume that η 1 has no long run impact on output, d 11 (j) = 0. j=0 The second row of the matrix equation (1) is Π(t) = log P(t) = d 21 (j)η 1 (t j) + d 22 (j)η 2 (t j). (2) j=0 j=0 The first term on the right hand side in equation (2), j=0 d 21(j)η 1 (t j), is the underlying core inflation. In the estimation, the VAR equation (1) is inverted into a moving average representation. Then the matrix D, η 1, and η 2 can be recovered from the estimation. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

15 Basic Concept of Core Inflation Core InflationFig. and 7. Measured the and Retail core inflation. Price -, Core; Index, RPI; in the RPIX U.K. Year Fig. 8. Measured and core inflation. -, Core; -, RPI; RPIX. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

16 Measuring Core Inflation CPIX CPIX A widely used method to measure core inflation is to exclude a fixed set of goods and services of volatile price movement from the CPI. A popular choice for the excluded items are food, energy, and indirect taxes. This index is often called CPIXFET. Studies have found that the CPIX is less efficient (in terms of standard deviation and RMSE) than other alternatives, even the CPI. A possible explanation is that excluded items such as food and gasoline can have long run impact on the economy. On the other hand, the remaining basket may contain price movements that are transitory. Currently the Bank of Canada CPIX excludes eight components in the CPI: fruits, vegetables, gasoline, fuel oil, natural gas, tobacco products, intercity transportation, and mortgage interest cost. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

17 Measuring Core Inflation Trimmed Mean Trimmed Mean From a statistical perspective, the CPI is a sample mean of price ratios, weighted by the expenditure shares. Our goal is to estimate the population mean of price ratios. It is well-known that the most efficient estimator of the population mean of a normal distribution is the sample mean of random experiments. But this is not necessary true for distributions with fat tails. In this case it can be more efficient to discard the samples (say 10% or 25%) from the tails on both sides. In the extreme case we can take the sample median as the estimator. Using data from the U.S. CPI, Cechchetti (1997) finds that the 10% trimmed mean is the most efficient estimator compared to the CPI, CPIXFET, the 25% trimmed mean, and the median. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

18 Measuring Core Inflation Variance-Weighted Price Index Variance-Weighted Price Index (CPIW) The Theil-Törnqvist index described above uses the expenditure shares s i as probability weights for the price ratios. Effectively, this assumes that the variance σi 2 of the distribution of price ratio of good i is inversely proportional to s i. But for the purpose of measuring inflation for monetary policy, this may not be the optimal choice. The price ratios should be weighted according to the strength of their signals-to-noise ratios. In the CPIX, we effectively assign zero weights to selected goods such as gasoline, food, etc. A more reasonable approach is to assign weights according to the inverse of σi 2, that is, 1/σi 2 si = N. i=1 1/σ2 i The resulting index is called a variance-weighted price index. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

19 Measuring Core Inflation Dynamic Factor Index Dynamic Factor Index The dynamic factor approach combines the cross-sectional variations of different markets with their time-series signals. The basic idea is that the measured inflation rate of each good or service πi t can be decomposed into a common trend component πc t and an idiosyncratic component xi t: π t i = Π t c + x t i, (3) where πi t = log(pi t/pt 1 i ). Notice that Π t c is the same for i = 1,..., N. The overall inflation rate is Π t = N s i log i=1 since N i=1 s i = 1. pt i p t 1 i = N s i (Π t c + xi t ) = Π t c + i=1 N s i xi t, (4) Kam Yu (Lakehead) 4 Statistical Approach Winter / 26 i=1

20 Measuring Core Inflation Dynamic Factor Index If the last term in equation (4), N i=1 s ix t i = 0, then the measured inflation Π t is equal to Π t c, the desired core inflation. For reasons discussed above, however, the term is generally not zero. Write equation (3) in vector form for all N goods and services, we have π t = Π t c1 + x t, where π t = (π t 1,..., πt N ), 1 = (1,..., 1), and x t = (x t 1,..., x t N ). The variables Π t c and x t are assumed to follow some stochastic processes. In particular, ψ(l)π t c = δ + ξ t, Θ(L)x t = η t, where ψ(l) is a polynomial of the lag operator L, Θ(L) is a polynomial matrix of L, and ξ t and η t are respectively scalar and vector i.i.d. processes. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

21 Measuring Core Inflation Dynamic Factor Index If ψ(l) = 1 and Θ(L) = I, the identity matrix, the system becomes static. Cecchetti (1997) assume that both variables follow an AR(2) process. That is, ψ(l) = 2 1 ψ 1 L ψ 2 L, { Θ ij (L) = 1 θ 1 L θ 2 L 2, if i = j, 0 otherwise. For further development of the dynamic factor approach see Reis and Watson (2010) and Stock and Watson (2016). Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

22 Evaluations and Discussions Comparison and Evaluations: Volatility The variety of methods and approaches are often compared and evaluated using a set of criteria. The prime objective of a core inflation is to eliminate the high frequency noises from the CPI. Volatility is evaluated by statistical efficiencies such as standard deviation and root mean squared error (RMSE). These statistics are calculated based on the deviation of the inflation from a centred (say 36 month) moving average of the inflation in the CPI. Many studies have found that the performance of the CPIX is unsatisfactory. Cecchetti (1997) points out that the RMSE criterion is symmetric on overestimating and underestimating inflation. But central bankers are more averse to underestimating when inflation is high, while more averse to overestimating when inflation is near zero. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

23 Evaluations and Discussions Comparison and Evaluations: Persistence After a shock, inflation may take time to return to its long run equilibrium. Persistence is a measure to capture the memory intensity of the shock. For example, consider an autoregressive stochastic process AR(p): y t = α + p φ j y t j + ɛ t. Let the sum of impulse coefficients be ρ = p j=1 φ j. Then the cumulative impulse response is defined as CIR = 1/(1 ρ). j=1 Both ρ and the CIR have been proposed as a measure of persistence. Using ρ as a measure, Khan et al (2015) find that the dynamic factor index outperforms other indices in measuring persistence in inflation dynamics. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

24 Evaluations and Discussions Discussions Unlike the economic approach, where the cost of living index is well-defined, there is no unifying concept in measuring core inflation. The Quah and Vahey (1995) definition of core inflation relies on the idea of monetary neutrality, which implies a vertical long-run Phillips curve. Wynne (2008) notes that this definition is problematic since the observed long-run Phillips curve is not vertical. Nevertheless, central bankers may still want to measure the component of inflation that has no long run impact on output. Many economists agree that the performance of CPIX is not satisfactory. A monthly index is too noisy for the purpose of conducting monetary policy. A quarterly index is recommended. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

25 Evaluations and Discussions The dynamic factor index is a pure statistical approach and devoid of economic meaning. Reis and Watson (2010), however, decompose the measured inflation into six fundamental shocks, which include anticipated money, unanticipated money, aggregate productivity, sectoral productivity, firm-level productivity, and anticipated tax changes. A problem of the approach is that every period when new data are available, the model has to be reestimate again. The inflation rate is therefore undergoing constant revision. This can be defended from the Bayesian perspective. We use all available information to estimate the core inflation, and update the estimation when new information arrives. Kam Yu (Lakehead) 4 Statistical Approach Winter / 26

26 Evaluations and Discussions Further Readings The dynamic factor index is described in Stephen G. Cecchetti (1997) Measuring Short-Run Inflation for Central Bankers, Federal Reserve Bank of St. Louis Review, 79(3), For a review of the VAR model, see James H. Stock and Mark W. Watson (2001), Vector Autoregressions, Journal of Economic Perspectives, 15(4), Kam Yu (Lakehead) 4 Statistical Approach Winter / 26