3.3 Dynamic panel quantile regression
3.3.2 The fitted value approach
proper-ties can be shown as
√
3.3.2 The fitted value approach
Lin (2010) solved the endogenous problems by using a fitted value and utilized the shrinkage method proposed by Koenker (2004) to eliminate an individual effect.
Consider the following model,
Qyit(τ |ηi, yit−1, xit) = ηi + α(τ )yit−1+ x0itβ(τ ). (13) Contrary to equation (11), individual effects ηi are fixed, which means ηi will not change with τ such as in Koenker (2004). yit is an dependent variable, yit−1 is a dynamic term, and xit is a covariate. α(τ ) and β(τ ) are interesting parameters.
Similarly, assume zit is a valid instrument variable, and we can estimate the parameters by a two-step procedure. First, run OLS of yit−1 on zit, and we can
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Second, we have to solve the following objective function for estimating parameters in equation (14): function, and ωkis a weight which controls the relative influence of the τkth quantile.Again, when λ → 0, the dynamic panel quantile regression estimators ( ˆαL(τ ), ˆβL(τ )) can be obtained. The fitted value approach of Lin (2010) is applied in my thesis, because a fixed effects model is a common choice for macroeconomists (Judson and Owen, 1999).
For estimating the variance-covariance matrix of ˆβ(τ ), bootstrapping is utilized here.7 Bootstrapping is a re-sampling method, which can help us obtain properties of an estimator from an approximating distribution. In practice, we could sample from observations {yi, xi, i = 1, . . . , n} according to i, and hence a new sub-sample {yi∗, x∗i} can be obtained. Then, we run a quantile regression of yi∗ on x∗i, and the estimator ˆβ∗(τ ) can be obtained. Next, we resample and run the regression as above, and we can get a number of ˆβ∗(τ, b), where b = 1, . . . , B and B is a number of re-sampling times. For example, if we do a re-sampling B times, we can obtain estimators ˆβ∗(τ, 1), ˆβ∗(τ, 2), . . . , ˆβ∗(τ, B). And then, the variance-covariance matrix
7Bootstrapping is proposed by Efron (1979). The application of bootstrapping to quantile regression, see Buchinsky (1995, 1998).
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This section is organized as follows. At first, data descriptions, sources and char-acteristics are summarized. And later on, average long term data is used in cross-sectional analysis. And finally, panel data is used to estimate empirical results which consist of baseline, extensive, country group-specific and period specific analysis.
4.1 Data
The main dataset consists of a panel of 91 countries (see Appendix A) from 1960 to 2006, and the main sources are the IMF’s International Financial Statistics (IFS) and the Penn World Table version 6.3 (PWT 6.3). Some gaps are filled with the World Bank’s World Development Indicators (WDI), Desai et al. (2003),8 Mitchell (2007a–c) and the United Nations’ National Accounts Statistics database.
Inflation is measured by the annual change rate in the consumer price index.
Fiscal deficits are nominal central government deficits scaled by narrow money stock (M1) and nominal GDP, so I calculated the deficit-to-money ratio as fiscal deficits over M1 and the deficit-to-GDP ratio as fiscal deficits over nominal GDP. The money growth rate is the annual change in the money stock (M1). The growth rate of real GDP per capita is the annual change in the real GDP per capita, which represents real economic growth. The oil price is the average crude price of petroleum in local currency, and oil price inflation is its annual change. The benefit of measuring the oil price in local currency is that each country could face various oil prices. Finally, openness is measured by the average of the import- and export-to-GDP ratio. (For detailed data sources and descriptions: see Appendix B.)
Table 1 provides a summary of the characteristics of the original data of selected countries over the period from 1960–2006 (panel A), 1970–2006 (panel B), 1980–2006 (panel C) and 1990–2006 (panel D). We can see that from 1960–2006 to 1990–2006,
8I thank Dr. Raj M. Desai for generously sharing their dataset.
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Table 1: Descriptive statistics of selected countries
mean quantile median quantile standard minimum maximum number of
0.25 0.75 deviation countries
(A) 1960–2006
inflation rate (%) 27.27 2.69 6.16 12.88 307.09 -100.00 10945.70 91
deficits/money (%) 23.83 5.67 18.08 33.93 39.35 -180.64 1056.96 91
deficits/GDP (%) 3.36 0.98 2.80 4.92 5.61 -22.24 204.56 91
money growth rate (%) 29.50 6.73 13.17 21.66 275.45 -99.90 11673.40 91
growth of real GDP per capita (%) 2.39 -0.01 2.47 4.92 5.87 -42.95 68.87 91
oil price inflation (%) 84.64 -0.54 4.87 24.36 3273.49 -63.42 213153.20 91
openness (%) 34.22 19.30 27.81 42.43 24.68 0.15 228.47 91
(B) 1970–2006
inflation rate (%) 31.05 3.29 7.44 14.44 336.16 -100.00 10945.70 96
deficits/money (%) 25.82 5.96 19.08 36.34 42.67 -180.64 1056.96 96
deficits/GDP (%) 3.63 1.04 2.95 5.28 6.00 -21.98 204.56 96
money growth rate (%) 31.19 7.15 14.11 23.35 274.36 -62.55 11673.40 96
growth of real GDP per capita (%) 2.20 -0.34 2.29 4.75 5.92 -41.11 60.37 96 oil price inflation (%) 101.68 -1.62 11.42 31.01 3591.80 -63.42 213153.20 96
openness (%) 36.72 21.08 30.04 46.49 25.01 0.15 228.47 96
(C) 1980-2006
inflation rate (%) 39.89 2.73 6.67 13.96 392.18 -100.00 10945.70 101
deficits/money (%) 26.92 5.62 19.17 38.57 48.69 -221.44 1056.96 101
deficits/GDP (%) 3.57 0.96 2.85 5.37 6.57 -22.66 204.56 101
money growth rate (%) 38.78 6.53 13.50 23.19 322.70 -62.55 11673.40 101
growth of real GDP per capita (%) 1.87 -0.45 2.08 4.37 5.39 -36.18 56.40 101 oil price inflation (%) 176.93 20.34 30.74 46.94 5789.44 -19.34 301488.60 101
openness (%) 38.23 22.11 31.97 49.84 25.26 0.15 228.47 101
(D) 1990-2006
inflation rate (%) 24.89 2.44 5.70 11.82 276.81 -13.85 7485.49 98
deficits/money (%) 23.66 3.33 15.47 35.00 44.10 -180.64 551.36 98
deficits/GDP (%) 2.83 0.62 2.38 4.54 6.51 -21.98 204.56 98
money growth rate (%) 27.45 6.80 13.32 23.09 230.26 -29.67 6724.82 98
growth of real GDP per capita (%) 2.26 0.08 2.38 4.60 5.14 -36.18 56.40 98
oil price inflation (%) 12.86 -6.43 10.24 32.10 24.84 -40.27 104.42 98
openness (%) 40.64 24.55 34.01 53.22 25.09 0.15 228.47 98
Source: the International Financial Statistics, Mitchell (2007a–c), the Penn World Table 6.3, Desai et al. (2003), the World Development Indicators and the United Nations’
National Accounts Statistics database.
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the average inflation rates are 27.27%, 31.05%, 39.89% and 24.89%, and the standard deviations of the inflation rate are 307.09, 336.16, 392.18, and 276.81 respectively.
Inflation tends to be higher and more volatile since 1960, and it becomes lower and more stable after 1990. Compared with the median of inflation rate, 6.16%, 7.44%, 6.67% and 5.70% respectively, the average inflation rates are higher. This means that the average is prone to be affected by extreme observations. Quantile regression can avoid the estimated outcomes affected by extreme observations.
The deficit-to-money ratio is similar. From 1960–2006 to 1990–2006, the average deficit-to-money ratios are 23.83%, 25.82%, 26.92% and 23.66%, and the standard deviations are 39.35, 42.67, 48.69 and 44.10 respectively. On the other hand, the average deficit-to-GDP ratios are 3.36%, 3.63%, 3.57% and 2.83%, and the standard deviations are 5.61, 6.00, 6.57 and 6.51 respectively. As we can see, both the deficit-to-money ratio and the deficit-to-GDP ratio tend to be larger and become smaller after 1990, and yet the volatility is still large after 1990.
The average money growth rates are 29.50%, 31.19%, 38.78% and 27.45% from 1960–2006 to 1990–2006 respectively, and the standard deviations are 275.45, 274.36, 322.70 and 230.26. Similarly, both the growth rates and volatility reach a peak in the 1980s, and decline after 1990. About the other controlled variables, the growth rate of the real GDP per capita declines until 1980–2006, and becomes rapid after 1990. The oil price inflation rises higher until reaching a peak during 1980–2006, and sharply drops in 1990–2006. Finally, openness is consistently growing higher, and volatility is stable.
Next, for comparing countries in various development levels, I classify country groups according to the income level and OECD membership which are based on the World Bank list of economies (July 2009) (see Appendix C). However, the clas-sification “OECD” consists of countries which are not only OECD members but also in a high-income level. Some OECD members in middle- or low-income levels are excluded such as Turkey, and the group of high-income countries contains OECD
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Table 2: Descriptive statistics of specific income groups (1960–2006)
mean quantile median quantile standard minimum maximum number of
0.25 0.75 deviation countries
(A) high-income countries
inflation rate (%) 7.02 2.15 3.95 7.71 16.22 -20.63 373.82 33
deficits/money (%) 19.65 3.16 12.47 28.16 38.14 -110.26 613.14 33
deficits/GDP (%) 2.73 0.67 2.22 4.55 4.49 -22.24 26.74 33
money growth rate (%) 12.90 5.43 10.02 15.42 23.02 -76.85 430.17 33
growth of real GDP per capita (%) 2.95 0.98 2.80 4.90 4.27 -21.97 27.81 33
oil price inflation (%) 15.34 -2.84 1.98 19.49 47.00 -63.42 405.31 33
openness (%) 40.91 23.27 33.98 49.08 30.09 4.63 228.47 33
(B) middle- and
low-income countries
inflation rate (%) 38.80 3.54 8.04 16.47 384.01 -100.00 10945.70 58
deficits/money (%) 26.20 7.91 20.26 36.13 39.84 -180.64 1056.96 58
deficits/GDP (%) 3.72 1.15 3.12 5.02 6.13 -21.98 204.56 58
money growth rate (%) 38.94 8.14 15.46 25.00 344.25 -99.90 11673.40 58
growth of real GDP per capita (%) 2.07 -0.78 2.09 4.96 6.59 -42.95 68.87 58
oil price inflation (%) 124.07 0.00 7.11 30.13 4099.92 -60.09 213153.20 58
openness (%) 30.42 17.42 25.10 37.61 20.03 0.15 122.23 58
Source: the International Financial Statistics, Mitchell (2007a–c), the Penn World Table 6.3, Desai et al. (2003), the World Development Indicators and the United Nations’
National Accounts Statistics database.
classification. Therefore, high-income or OECD countries represent economies in higher development, and middle- and low-income or non-OECD countries represent developing economies.
Table 2 is a summary of the characteristics of the original data in a high-income country group (panel A) and a middle- and low-income country group (panel B). We can see that in high-income countries, the average inflation is 7.02% and its standard deviation is 16.22. In middle- and low-income countries, the average inflation rate is 38.80% and 384.01. Accordingly, inflation is lower and more stable in high-income countries.
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Table 3: Descriptive statistics of OECD and non-OECD countries (1960–2006)
mean quantile median quantile standard minimum maximum number of
0.25 0.75 deviation countries
(A) OECD countries
inflation rate (%) 6.36 2.33 4.12 7.97 7.06 -13.85 84.22 24
deficits/money (%) 14.99 2.80 10.56 22.81 22.38 -72.42 160.32 24
deficits/GDP (%) 2.56 0.63 2.01 4.12 3.95 -22.24 20.79 24
money growth rate (%) 11.59 5.60 9.47 15.28 12.87 -62.55 192.09 24
growth of real GDP per capita (%) 2.90 1.16 2.74 4.61 3.27 -13.56 21.37 24
oil price inflation (%) 14.65 -2.49 2.70 19.42 44.95 -63.42 292.31 24
openness (%) 30.02 20.79 28.28 36.24 14.64 4.63 92.15 24
(B) non-OECD countries
inflation rate (%) 34.76 3.03 7.31 15.15 357.58 -100.00 10945.70 67
deficits/money (%) 26.99 7.72 20.36 37.50 43.43 -180.64 1056.96 67
deficits/GDP (%) 3.65 1.14 3.10 5.08 6.07 -21.98 204.56 67
money growth rate (%) 35.91 7.61 14.63 24.27 320.69 -99.90 11673.40 67
growth of real GDP per capita (%) 2.21 -0.64 2.24 5.11 6.54 -42.95 68.87 67 oil price inflation (%) 109.71 -0.07 5.96 28.44 3814.75 -60.09 213153.20 67
openness (%) 35.73 18.74 27.59 46.28 27.24 0.15 228.47 67
Source: the International Financial Statistics, Mitchell (2007a–c), the Penn World Table 6.3, Desai et al. (2003), the World Development Indicators and the United Nations’
National Accounts Statistics database.
Scaling by money stock, the average and standard deviation of the deficit-to-money ratio is 19.65% and 38.14 in high-income countries, and 26.20% and 39.84 in middle- and low-income countries. Then, scaling by GDP, the average and standard deviation of the deficit-to-GDP ratio are 2.73% and 4.49 in high-income countries, and 3.72% and 6.13 in middle- and low-income countries. Obviously, whether scaling by money or GDP, the fiscal deficit is more critical in middle- and low-income countries.
The average money growth rate is 12.90% in high-income countries and 38.94%
in middle- and low-income countries. Its standard deviation is 23.02 in high-income
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countries and an astounding 344.25 in middle- and low-income countries. Appar-ently, money growth gets better control in high-income countries. The average growth rate of real GDP per capita and its standard deviation are 2.95% and 4.27 in high-income countries, and 2.07% and 6.59 in middle- and low-income countries.
The long-term economic growth is higher and more stable in high-income countries.
The average oil price inflation is 15.34% in high-income countries, but is 124.07%
in middle- and low-income countries. It might be because exchange rates devalu-ate in middle- and low-income countries gredevalu-ater than in high-income countries. And finally, high-income countries are more open and have a higher dependence on trade.
Table 3 provides a summary of the descriptive statistics of the original data in OECD countries (panel A) and non-OECD countries (panel B). We can see that the characteristics of variables do not change a lot. The average inflation rate and its standard deviation are 6.36% and 7.06 in OECD countries, and 34.76% and 357.58 in non-OECD countries. This means that inflation is at a higher level and more volatile in non-OECD countries.
The average fiscal deficits scaling by money are 14.99% and 26.99% in OECD and non-OECD countries respectively, and the standard deviations are 22.38 and 43.43. Scaling by GDP, the average fiscal deficits are 2.56% and 3.65% in OECD and non-OECD countries, and the standard deviations are 3.95 and 6.07 respectively.
Similarly, a fiscal deficit is a more critical problem in non-OECD countries.
About other variables, the average money growth rate is 11.59% in OECD coun-tries and 35.91% in non-OECD councoun-tries and its standard deviation is 12.87 in high-income countries and 320.69 in non-OECD countries. Money growth is well controlled in non-OECD countries. The average growth rate of real GDP per capita and its standard deviation are 2.90% and 3.27 in OECD countries, and 2.21% and 6.54 in non-OECD countries. Similarly, the long-term growth rate is higher and more stable in OECD countries. The average oil price inflation is 14.65% in OECD countries and 109.71% in non-OECD countries. And finally, openness in OECD
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countries is higher than in non-OECD countries, and this means that OECD coun-tries are more dependent on trade.
In general, we can find that from 1960–2006 to 1980–2006, the macroeconomic performance and related variables deteriorate and become more volatile, and they develop into better and more stable entities after 1990. Openness is the only vari-able that consistently goes higher. On the other hand, whether classifying countries by income level or OECD membership, macroeconomic variables, including inflation and fiscal deficits, perform better and are more stable in countries in higher develop-ment, and perform worse and are more unstable in countries in lower development.