Econometrics On Convergence & Cross-sectional Dependenced.sul/book/Convergence... · 2019-10-16 ·...

Econometrics On Convergence & Cross-sectional Dependence

October 18, 2019

Donggyu Sul

University of Texas at Dallas

• Econometric Theory on Convergence

1. Phillips and Sul (2007) “Transition modeling and econometric convergence tests”, Econometrica

2. Kong, Phillips and Sul (2019) “Weak sigma convergence: Theory and Applications”, JoE, forthcoming

• Econometric Theory on Identification of Unknown Common Factors

1. Parker and Sul (2016) “Identification of unknown common factors: Leaders and followers”, JBES

2. Kwak and Sul (2019) “A new identification procedure for unknown integrated common factors with a convergent panel data”,

• General & Application

1. Sul (2019), Panel data econometrics: Common factor analysis for empirical researchers, Taylor & Routledge

2. Greenaway-McGrevy, Mark, Sul and Wu (2018), “Identifying exchange rates common factors”, IER

Key Reference

• Two-way FE Panel Regression - See Sul (2019 Chapter 6)

Issues: Two-way FE Regression

𝑦𝑖𝑡 = 𝑎𝑖 + 𝜃𝑡 + 𝛽𝑥𝑖𝑡 + 𝑢𝑖𝑡




𝑦𝑖𝑡 = 𝛽 𝑥𝑖𝑡 + 𝑢𝑖𝑡





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WG transformation:

Eliminating individual mean and

common factor

What’s left-over then?




Example: Log Infant Mortality Rates

• What do you see?

• Want to explain co-movements?

• Possibly determinants?

• Do you think you can do with the two-way FE

regression?




𝑦𝑖𝑡 𝑦𝑖𝑡


• Alternative Solution? • Do Not eliminate unknown common factor

• But Do include potential factors

• How to find potential factors, then?1. How many of them?

2. What are they?

3. Is it easy to do?



𝑦𝑖𝑡 = 𝑎𝑖 + 𝛿𝑖′𝐺𝑡 + 𝛽𝑥𝑖𝑡 + 𝑢𝑖𝑡


• Alternative Solution? • Do Not eliminate unknown common factor

• But Do include potential factors

• How to find potential factors, then?1. How many of them?

2. What are they?

3. Is it easy to do?



𝑦𝑖𝑡 = 𝑎𝑖 + 𝛿𝑖′𝐺𝑡 + 𝛽𝑥𝑖𝑡 + 𝑢𝑖𝑡

Research Topic of today’s talk

Data Types & Methods

𝑦𝑖𝑡 = 𝑎𝑖 + 𝛿𝑖′𝐺𝑡−1 + 𝛽𝑥𝑖𝑡−1 + 𝑢𝑖𝑡

An Example First:

Greenaway et al (2018) identified the US dollar and Euro values as potential factors.

• US dollar value is approximated by taking 27 bilateral spot rates with the US dollar numeraire

• Euro value is approximated in the similar way

• Run the below predictive regression (all data are logged and then first differenced)

• Showed that two identified factors improve forecasting performance A LOT!

• Surprisingly economic fundamentals are not much helpful in terms of forecasting.

Greenaway et al (2018) used stationary data: Has to use Parker and Sul (2016) method.

With nonstationary data (or trended stationary data) with weak sigma convergence:

One can use Kwak and Sul (2019) method.

Data Types & Methods

Data preparation:

1. Balanced panel data (T x n): No household level data.

• Household survey data: Most disaggregate data. Little cross-sectional dependence

• Very hard to estimate common factors

• More aggregated level such as city, firm, state, nation is better: Easy to estimate the common

behavior

• Example: John Smith in LA is not related to another John Smith in New York (in terms of crime

activity), but the crime rate in LA is much similar to that in New York. Why? See Sul (2019, Chap 4)

2. At least T > 20, n > 20. Too small T => Can’t do anything. More n is better

3. Collect potential factors (T x k): k national aggregate data. Not panel.

4. Check stationarity (with cross-sectional average).

• Take a log first. Run ADF test (not first differenced), or just plot it.

• As long as a panel data has trending behavior, treat as if they were non-stationary.

• Level is different from Growth: Do not inter-change the economic meaning: A poor country needs

to grow faster to catch up with a rich country => If so, level convergence, but growth divergence.

Flow Chart for Methods

Cross-sectional

mean:

Nonstationary or

has a trend

No

Run Weak sigma convergence TestKong, Phillips and Sul (2019)

No Phillips & Sul (2007)Automatic Clustering

analysis:Find sub-convergent

clubs and analyze club characteristics,

Phillips & Sul (2019)In progress..

Co-divergence: Finding the source of

Divergence

Yes

Kwak and Sul (2019): Identify unknown common factors using weak sigma test.

Yes

Parker and Sul (2016): Identify unknown common factors using factor number estimation

Cross-Sectional Dependence

Real World Data: All cross-sectionally dependent

Fact: More highly aggregated data, more cross-sectionally dependent (Sul, 2019 Chap 4)

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-0.2

0

0.2

0.4

0.6

0.8

1

6 7 8 9 10 11 12 13 14

log average populations

Sam

ple

Corr

elati

ons

5% 95%

Figure 4.4 from Sul (2019): Shows the lower 5% and upper 95% cross-sectional correlations after sorting log difference in personal income (per capita) in3,089 counties in terms of population.

High population -> more aggregated -> High cross-sectional correlation

Another example: Crime rates, State crime rates is less correlated compared with metropolitan crime rates. Why?


Cross-sectional

mean:

Nonstationary or

has a trend

No

Run Weak sigma convergence TestNo Phillips & Sul (2017)

Automatic Clustering analysis:

Find sub-convergent clubs and analyze

club characteristics,



Divergence

Yes


Yes


Static Factor Model

𝑦𝑖𝑡 = 𝑎𝑖 + 𝜆𝑖′𝐹𝑡 + 𝑦𝑖𝑡

𝑜

• Economic meaning: See Sul (2019, Chap 1)o 𝐹𝑡: a vector of common factorso 𝜆𝑖: economic distance between 𝑦𝑖𝑡 and the common factors. o Ex: 𝑦𝑖𝑡 = earning. 𝐹𝑡= common training or event. 𝜆𝑖=individual productivity. 𝑦𝑖𝑡

𝑜 =pure idiosyncratic earning

• Factor number unknown. Need to estimateo Recommendation: Bai and Ng (2002)’s IC2 criterion. Also See Sul (2019, Chap 4) for other criteriao Factor estimation: Single factor -> cross-sectional average, more than single factors -> Principal

Component (PC) estimation. o STATA, MATLAB, GAUSS codes are available (Visit

https://personal.utdallas.edu/~dxs093000/book/panel.htm)

https://personal.utdallas.edu/~dxs093000/book/panel.htm

Parker and Sul (2016)

Difference between Static and Economic Factors:

𝑦𝑖𝑡 = 𝑎𝑖 + 𝛽1𝑖𝐺1𝑡 + 𝛽2𝑖𝐺2𝑡 + 𝑦𝑖𝑡𝑜 ,

where 𝐺1𝑡 and 𝐺2𝑡 are correlated each other. They are “Economic” factors.

10.9

0.8 0.8

0.9

0.7

AB

C

D

EF

Economic Factors are possibly correlated each other.

However statistically it is impossible to identify both factor loadings and dependent factors. By using an orthogonal transformation (ex. Cholesky decomposition), we can get “statistical factors”

𝐺1𝑡

𝐺2𝑡=

𝑎11 𝑎12

0 𝑎22

𝐹1𝑡

𝐹2𝑡

where 𝐹1𝑡 is independent from 𝐹2𝑡.


If 𝑃1𝑡 and 𝑃2𝑡 are true factors, then the following regression residuals should not have any significant factor.

𝑦𝑖𝑡 = 𝑎𝑖 + 𝛽1𝑖𝑃1𝑡 + 𝛽2𝑖𝑃2𝑡 + 𝑢𝑖𝑡

If 𝑃1𝑡 = 𝐺1𝑡 but 𝑃2𝑡 ≠ 𝐺2𝑡, then the regression residuals must have a single true factor.

When the number of common factors is single, then Parker and Sul (2016) considers a dominant leadership model.

10.9

0.8

0.7

AB

C

D

EF

Run the following regression:

𝑦𝑖𝑡 = 𝑎𝑖 + 𝛽1𝑖𝑦1𝑡 + 𝑢𝑖𝑡,

If 𝑦𝑖𝑡 is the dominant leader, then the regression residualshave zero factor.

EX: Gaibulloev et al (2013). Use transnational terrorism data,find a single factor by using Bai Ng’s IC2, and identifyLebanon’s terrorism is the common factor to the rest.


Remarks:1. Parker and Sul allow somewhat mild deviations between 𝑃𝑡 and 𝐺𝑡

2. To estimate the number of common factors, the data should be stationary. Also must standardized (divided by each time series standard deviation)

3. Assume 𝐹𝑡 = 𝐹𝑡−1 + 𝑣𝑡, Δ𝑦𝑖𝑡 = 𝜆𝑖′Δ𝐹𝑡 + 𝑦𝑖𝑡

𝑜 − 𝑦𝑖𝑡−1𝑜 . If 𝑦𝑖𝑡

𝑜 is i.i.d. then noise becomes twice but signal becomes weakened.

Standardized log Infant Mortality Rate First differenced and Standardized Series


Cross-sectional

mean:

Nonstationary or

has a trend

No







Divergence

Yes


Yes


Phillips and Sul (2007)’s Relative Convergence

𝑦1𝑡 is relatively converging to 𝑦2𝑡 iff

lim𝑡→∞

𝑦1𝑡/𝑦2𝑡 = 1

𝑦𝑖𝑡 is relatively converging to it’s cross-sectional mean iff

lim𝑡→∞

𝑦𝑖𝑡/𝑦𝑛𝑡 = 1 for all i

The notion of the relative convergence was born for accurate testing for convergence.

Note that the so-called 𝛽 −convergence always hold as long as a panel data is stationary. Opposite is not true. See Sul (2019, Chap 7 for more details)

Phillips and Sul’s Relative Transition Curve

Time Varying Factor loadings:

𝑦𝑖𝑡 = 𝑏𝑖𝑡∗ 𝜃𝑡 + 𝑦𝑖𝑡

𝑜 = 𝑏𝑖𝑡∗ +

𝑦𝑖𝑡𝑜

𝜃𝑡𝜃𝑡 = 𝑏𝑖𝑡𝜃𝑡

Impossible to identify 𝑏𝑖𝑡 since the number of unknown is larger than the number of observations.Use the following relative transition curve to approximate 𝑏𝑖𝑡:

ℎ𝑖𝑡 =𝑦𝑖𝑡

𝑦𝑛𝑡=

𝑏𝑖𝑡

𝑏𝑛𝑡

where 𝑦𝑛𝑡 is the cross-sectional average of 𝑦𝑖𝑡 .

The cross-sectional variance of ℎ𝑖𝑡 is given by

𝐻𝑡 = 𝑛−1 ℎ𝑖𝑡 − 1 2

Then the logt regression can be used for the relative convergence

𝑙𝑛𝐻1

𝐻𝑡− ln 𝑙𝑛𝑡 = 𝑎 + 𝛽𝑙𝑛𝑡 + 𝑢𝑡 for t = r, … , T

where r = int(0.3T). The initial 30% sample is needed to discard.

Phillips and Sul’s Relative Convergence

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Phillips and Sul’s Automatic Clustering Algorithm

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If relative convergence is rejected, then seek for the possibility of sub-club convergence.

Automatic clustering mechanism: See Sul (2019, Chap 7)STATA, R, and GAUSS codes are available.

Next, find out the club characteristics by running multi-nomial logit regression.

Define 𝑆𝑖 as the club-membership. Then

Pr[𝑆𝑖 = 𝑐] = exp 𝛽𝑐𝑋𝑖 /∑exp(𝛽𝑘𝑋𝑖)

This method has been popularly used. Not identifying a sub-common factor, but finding determinants of sub-club memberships.


Cross-sectional

mean:

Nonstationary or

has a trend

No







Divergence

Yes


Yes


Kong, Phillips and Sul (2019)’s Weak Sigma Convergence

• Phillips and Sul’s log t regression works when 𝑦𝑖𝑡 has a (non)stochastic trend. If not, the notion of the relative convergence does not work.

• In other words, if 𝑦𝑖𝑡 is stationary (or a growth rate), then a different notion of convergence is needed.

• Secrist (1933) and Barro and Sala-i-Martin (1992): (𝛽-convergence): initially poor countries grow faster than initially rich countries• Criticized by Hotelling (1933) and Friedman (1992): Statistical illusion.• See Sul (2019, Chap 7): Even when the cross-sectional variance of 𝑦𝑖𝑡 increases, 𝛽-convergence holds.

• Original notion of convergence: (Weak sigma convergence) If a cross-sectional variance is overall decreasing over time, weak sigma convergence holds.

• Traditionally many empirical researchers have used this notion of weak sigma convergence. Just they didn’t know how to test properly.

Sigma Convergence v.s. Beta Convergence

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-2

-1

0

1

2

3

0 5 10 15 20 25 30

• 𝛽-convergence: Secrist (1933), Barro and Sala-i-Martin (1992)

• A poor country needs to grow faster to catch up with a rich country

• Sounds reasonable, but a test becomes problematic

(𝑦𝑖𝑇−𝑦𝑖1)/(𝑇 − 1) = 𝑎 + 𝛽𝑦𝑖1 + 𝑢𝑖

• 𝛽 < 0 𝑦𝑖𝑡 is stationary (with nonstationary initial condition)

• Not related with convergence at all

• Example: Hotelling (1933), Select three series (stationary). Cross-sectional variance is stable.

Sigma Convergence v.s. Beta Convergence

• Friedman (1992): Criticize Barro and Sala-i-Martin (1992)• Surprisingly, many researchers still use the notion of 𝛽 convergence!• True convergence: Cross-sectional dispersion should be decreased• Sample cross-sectional variance measures the average distance between 𝑦𝑖𝑡 and its cross-sectional

mean:

• If 𝐾𝑛𝑡𝑦

converges to a constant, then 𝑦𝑖𝑡 is weakly 𝜎-converging to its cross-sectional mean.

• If 𝑦𝑖𝑡 has a static factor, then eliminate them before testing weak sigma convergence.

• If 𝑧𝑖𝑡 is weakly 𝜎-converging, then 𝑦𝑖𝑡 is weakly 𝜎-converging to its cross-sectional average of common components: ҧ𝜆′𝐹𝑡

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Kong, Phillips and Sul (2019)’s Weak Sigma Convergence

• KPS shows that the following simple trend regression can be used for weak sigma convergence𝐾𝑛𝑡 = 𝑎 + 𝜑𝑡 + 𝑢𝑡

where 𝐾𝑛𝑡 is the sample cross-sectional variance.

• Should use Newey-West HAC estimation for the proper t-statistic with the lag length of int[𝑇1/3].

• Interestingly, Sul (2019, Chap 7) shows that with (non)stochastic trend data, almost always (why?)

• Weak sigma convergence requires many n (cross-sectional units). T can be small (not too small).

• Weak sigma convergence can’t be used for automatic clustering (sub-convergence analysis)

• But it can be used for identifying unknown common factors (with (non)stochastic trended data).

Weak sigma convergence Relative convergence

Example

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3 4 5 6 7 8 9

log initial expenditure

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KLIPS (Korea Labor Income Panel Survey)17 years over more than 2,000 households.

Figure 7-3 from Sul (2019)

Table 7-1: Various Convergence Tests

Convergence Case t Convergence

All -0.033 -21.84 Yes

Di 0. 2 -0.034 -18.79 Yes

Rel. Convergence b tb

-0.831 -39.84 No

Weak Convergence t

n.a. 0.013 9.949 No

Kwak and Sul (2019)’s Identification Method

• Assume 𝑦𝑖𝑡 = 𝑐𝑖 + 𝜆𝑖′𝐹𝑡 + 𝑦𝑖𝑡

𝑜 . Further assume that 𝐹𝑡 has a (non)stochastic trend. Then the weak sigma convergence test is usually done with 𝑧𝑖𝑡 = 𝑐𝑖 + 𝑦𝑖𝑡

𝑜

• Suppose that 𝐹𝑡 = 𝜑′𝐺𝑡 + 𝑢𝑡 where 𝑢𝑡 = 𝑢𝑜,𝑡𝑇𝛾

2−

1

2 with 0 ≤ 𝛾 < 1 and 𝑢𝑜,𝑡 has a finite variance.

Then 𝐺𝑡 can be a long run determinant of 𝐹𝑡 since as 𝑇 → ∞, 𝜑′ 𝐺𝑡 → 𝐹𝑡.• Kwak and Sul show that the weak sigma convergence test can be used for identifying 𝐹𝑡.• Run the following regression:

𝑦𝑖𝑡 = 𝑐𝑖 + 𝛿𝑖′𝐺𝑡 + 𝑒𝑖𝑡 for each 𝑖

Ƹ𝑧𝑖𝑡 = Ƹ𝑐𝑖 + Ƹ𝑒𝑖𝑡

𝐾𝑛𝑡Ƹ𝑧 = 𝑎 + 𝜑𝑡 + 𝑢𝑡

If ො𝜑 < 0 significantly, then 𝐺𝑡 are determinants of 𝐹𝑡

Empirical Example

46 disaggregate PCE price indexes from 1978 to 2016 (from KPS’ example)Find one factor by using BN’s IC2

• After taking growth rate, another first difference, and standardization, See Sul (2019, Chap 4)• Before eliminating the common factor, the t-ratio becomes -3.67 (factor loadings are similar)• After eliminating the common factor, the t-ratio becomes -4.76

Table 1: (Non)Stochastic Trends in Inflation and Potential Determinants

Trend ADF (Trend) ADF (No Trend) Relevance

t Yes -3.989 n.a

Potential Determinant

FFR Yes -3.629 n.a. Yes

ln(GDP) Yes -1.549 n.a. Yes

lnIM Yes -1.757 n.a. Yes

lnGDP No n.a. -3.560 No

lnIM No n.a. -3.626 No

Empirical Example

• Use Federal Fund Rate as a potential factor:

• t-ratio for 𝜑 = -4.28

• FFR becomes the common factor to all 46 PCE inflations

• Parker and Sul (2016) method: Can’t identify FFR as unknown common factor to PCE inflations

Empirical Example

When a wrong variable is included:

Total import & Real GDP as examples.

Both explain the trend of the headline inflation well.

However… weak sigma convergence does not hold with both variables.

Empirical Example

When an irrelevant variable is included:

Real GDP growth rate: Stationary

A stationary variable cannot explain a nonstationary trend

Weak sigma convergence holds but meaningless!

Conclusion

• Cross-sectional dependence is useful information

• Do not eliminate, but utilize to find hidden but core determinant variable(s)

• Study how to identify unknown common factors by using various methods

• Important lessons:1. Do not just run two-way FE regression2. Do identify unknown common factors

Econometrics On Convergence & Cross-sectional Dependenced.sul/book/Convergence... · 2019-10-16 ·...

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