Panel Threshold Model

Transcription

1 Panel Threshold Model 黃河泉淡江大學財務金融系 June 5 7, 2016 Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

2 1 Linear Model 2 Threshold Model 3 Panel Threshold Model Single-Threshold Model Multiple-Threshold Model 4 Applications Investment and Financing Constraints Inequality on Growth Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

3 First Example: I Consider the following example: g i = β 0 + β 1 inf i + e i, i = 1, 2,, n (1) where the coefficient of main interest is β 1, i.e., the effect of inflation (inf) on economic growth (g). Note that, at this moment, we are considering n cross-country (hence cross-sectional, not panel) data. Under certain assumptions, the above model can also be applied to time series data (t = 1, 2,, T ). We will get back to panel data (i = 1, 2,, n; t = 1, 2,, T ) later. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

4 First Example: II In practice, the coefficient estimate of β 1 can either be: (significantly) positive, which means higher inflation is associated with faster growth, (significantly) negative, which indicates higher inflation leads to slower growth, or (insignificantly different from) zero, which suggests that inflation is unrelated to growth. Clearly, the commonly-used linear regression can account for one, and only one, possible relationship between variables of interest (growth and inflation, among others). Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

5 First Example: III Arguably, in low levels of inflation, we would expect the impact of inflation on growth is negligible. g i = β 0 + β 1 inf i + e i (2) (0) i.e., the estimate of β 1 is statistically insignificantly different from 0. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

6 First Example: IV In contrast, it is commonly found that (very) high inflation leads to slower growth, i.e., g i = β 0 + β 1 inf i + e i (3) ( ) i.e., the estimate of β 1 is significantly negative ( ). So, how can we accommodate these two (contrasting) possibilities in a (special) regression, i.e., a negligible link between growth and inflation when inflations are low but a negative association when inflations are high? Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

7 Second/Third Example I Chen and Lee (2005) assess the validity of the Armey curve, which posits that there is a non-linear relationship between government size and economic growth. 1 S.-T. Chen, C.-C. Lee / Journal of Policy Modeling 27 (2005) Fig. 1. Armey curve. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

8 Second/Third Example II A conventional (linear/nonlinear?) specification to this end is: g i = β 0 + β 1 gov i + β 2 gov 2 i + e i (4) (+) ( ) which is easy to estimate/interpret. A finding of (statistically significant) positive β1 and negative β 2 is taken as supportive evidence of the Armey curve. Notice the joint test of β 1 > 0 and β 2 < 0. 2 The quadratic specification in (4) is commonly used in the literature to identify whether the relationship between the dependent variable y and the explanatory variable x are U- or inverted-u-shaped. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

9 Second/Third Example III In a similar vein, our third example (in corporate finance) is to investigate whether if exists an optimal debt ratio (leverage, target capital structure). The implication of this theory is that firm s value reaches its maximum as long as its capital structure remains optimal. Thus, as long as deviating from the optimal capital structure, the value of the firm decreases accordingly. An inverted-u relationship between a firm s value and its leverage will be consistent with the theoretical prediction of target capital structure. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

10 Second/Third Example IV The quadratic framework can be applied as well: value i = β 0 + β 1 debt i + β 2 debt 2 i + e i (5) (+) ( ) where value denotes firm s value (say, return on equity), and debt is the debt as a ratio of total asset (leverage), i.e., a measure of capital structure. Again, if the hypothesis of target capital structure is valid, we would expect to find a (jointly, significantly) positive β 1 and negative β 2. 1 Chen, S. T. and Lee, C. C. (2005), Government size and economic growth in Taiwan: A threshold regression approach. Journal of Policy Modeling, 27, Lind, J. T. and Mehlum, H. (2010), With or without U? The appropriate test for a U-shaped relationship. Oxford Bulletin of Economics and Statistics, 72, Please check out the Stata command utest for implementation. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

11 Application A potential application: Ferreira et al. (2011) test the hypothesis that stock price informativeness (SPI) affects the structure of corporate boards (ratio of independent directors, BIndep). 3 They find a negative relation between price informativeness and board independence, i.e., stock price informativeness and board monitoring/independence are substitutes. Their results suggest that, if stock prices are informative, stock markets are able to perform a monitoring role like the one normally associated with the board of directors. However, it might be (substitutes/complements): Worth a try using China data. BIndep = β 0 + β 1 SPI + β 2 SPI 2 + e (+) ( ) 3 Ferreira, D., Ferreira, M. A., and Raposo, C. C. (2011), Board structure and price informativeness. Journal Financial Economics, 99, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

12 Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

13 Nonlinear/Threshold Regression I In order to allow for contrasting/different effects of inflation on economic growth in the low-inflation and high-inflation regime, respectively, we can rely upon the threshold regression of Hansen (2000): 4 { β10 + β gi = 11 inf i + e i, if inf i < γ β 20 + β 21 inf i + e i, if inf i γ where, inf i is the threshold variable (dividing all the observations into two groups), and γ is the unknown threshold value (to be estimated by the least squares approach). (6) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

14 Nonlinear/Threshold Regression II Note that, in this case, inflation not only is the explanatory variable, but also serves as the threshold variable. In practice, the threshold variable is not necessarily included in the list of explanatory variables. In addition to stationarity (time series data), all the explanatory and threshold variables must be exogenous. If prediction of the first example is correct, then we will find a statistically insignificant β 11 (inf i < γ, low-inflation regime), and a significantly negative estimate of β 21 (inf i γ, high-inflation regime). Therefore, a threshold regression model can accommodate different links (in terms of signs, magnitudes, and significances) between/among variables in distinct regimes. 4 Hansen, B. E. (2000), Sample splitting and threshold estimation. Econometrica, 68, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

15 Estimation I Equation (6) can be re-written in a general form as: { β y i = 1 x i + e i, if q i < γ β 2 x i + e i, if q i γ (7) where y = g, x = (1, inf), β 1 = (β 10, β 11 ), β 2 = (β 20, β 21 ), e (0, σ 2 ) and the threshold variable q = inf. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

16 Estimation II Equation (7) can be re-expressed as: y i = β 1x i I(q i < γ) + β 2x i I(q i γ) + e i = β x i (γ) + e i (8) where I( ) is an indicator function, β = (β 1, β 2 ), and x i (γ) = [ xi I(q i < γ) x i I(q i γ) Least squares estimation: The sum of squared residuals from equation (8) is: ] S 1 (γ) = ê(γ) ê(γ) (9) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

17 Estimation III The estimated threshold value is given as: The variance of the residual is: ˆγ = arg min γ S 1 (γ) (10) ˆσ 2 = 1 nê(γ) ê(γ) = 1 n S 1(ˆγ) (11) Once ˆγ is obtained, the vector of parameter estimates are: ˆβ = ˆβ(ˆγ) = [ ˆβ1 (ˆγ) ˆβ 2 (ˆγ) ] Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

18 Estimation IV First: Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

19 Estimation V Second: Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

20 Estimation VI Third: Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

21 Inference I In addition to estimation: Are we sure that there are (only) two regimes? Or, the linear (one regime) specification is just fine! i.e., the inferential problem. The (nonlinearity/threshold effect) hypothesis (see (7)) is: H 0 : β 1 = β 2 H 1 : β 1 β 2 Notice that: βs are vectors of parameters. Under the null hypothesis, the linear regression is fine. If we can reject the null hypothesis (of linearity), then we have a (two-regime, nonlinear) threshold regression. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

22 Inference II Let S 0 and S 1 be the residual sum of squares under the null hypothesis: y i = β 1x i + e i with residual ẽ i, and under the alternative hypothesis: { β y i = 1 x i + e i, if q i < γ β 2 x i + e i, if q i γ with residual ê i, respectively. Thus, the likelihood ratio test of H 0 is based on: F 1 = S 0 S 1 (ˆγ) ˆσ 2 = S 0 S 1 (ˆγ) S 1 (ˆγ)/n Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

23 Inference III Problem: The nuisance parameter issue (which one?), rendering the asymptotic distribution of F 1 non-standard, and strictly dominates the χ 2 distribution. Davies, R. B. (1977), Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64, Davies, R. B. (1987), Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

24 Inference IV The bootstrap procedure: Treat the regressors xi and threshold variable q i as given, holding their values fixed in repeated bootstrap samples. Take the regression residuals êi (from the threshold regression) as the empirical distribution to be used for bootstrapping. Draw (with replacement) a sample of size n from the empirical distribution and use these errors to create a bootstrap sample under H 0, i.e., ŷ i = ˆβ 1x i + ê i Notice that the test statistic F 1 does not depend on the parameter β 1 under H 0, so any value of β 1 may be used. Using the bootstrap sample (ŷi, x i, q i ), estimate the model under the null and alternative, and calculate the bootstrap value of the likelihood ratio statistic F 1. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

25 Inference V Repeat this procedure a large number of times and calculate the percentage of draws for which the simulated statistic exceeds the actual. This is the bootstrap estimate of the asymptotic p-value for F 1 under H 0. Hansen (1996) shows that this bootstrap analog produces asymptotically correct p-values. 5 The null of no threshold effect is rejected if the p-value is smaller than the desired critical value (1%, 5%, or 10%). 5 Hansen, B. E. (1996), Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica, 64, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

26 Application I Durlauf and Johnson (1995) suggest that cross-section growth behavior may be determined by initial conditions. 6 Hansen (2000) examines the issue by specifying: ln (Y/L) i,1985 ln (Y/L) i,1960 = α + β ln (Y/L) i, π 1 ln (I/Y ) i +π 2 ln(n i + g + δ) + π 3 ln(school) i + e i (12) where for each country i: (Y/L) i,t = real GDP per member of the population aged in year t; (I/Y )i = investment to GDP ratio; n i = growth rate of the working-age population; Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

27 Application II SCHOOL = fraction of working-age population enrolled in secondary school. The variables not indexed by t are annual averages over the period Following Durlauf-Johnson, Hansen sets g + δ = Durlauf-Johnson estimate (12) for four regimes selected via a regression tree using two possible threshold variables that measure initial endowment: per capita output Y/L, and the adult literacy rate LR, both measured in Please visit Hansen s website for lots of resources. Code/data: thr-hansen00etrica.do/dta. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

28 Application III Note that another interesting paper is: Hansen, B. E. (2016), Regression kink with an unknown threshold. Forthcoming in Journal of Business & Economic Statistics. The model is: y t = β 1 (x t γ) + β 2 (x t γ) + + β 3z t + e t where (a) = min[a, 0] and (a) + = max[a, 0] denote the negative part and positive part of a real number a. In the above model (say, y is firm value), the slope with respect to the variable x t (leverage, capital structure) equals β 1 for values of x t less than γ, and equals β 2 for values of x t greater than γ, yet the regression function is continuous in all variables. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

29 Application IV GDP Growth Rate ε n = 0.5n 0.5 ε n = 1n 0.5 ε n = 2n 0.5 ε n = 4n Debt/GDP 6 Estimated Durlauf, S. N. Regression and Johnson, P. A. Kink (1995), Multiple Function regimes and and cross-country 90% growth Numerical behavior. Journal Delta of Applied Meth Econometrics 10, Please also see Mankiw, N. G., Romer, D. and Weil, D. N. (1992), A contribution to the empirics of economic growth. Quarterly Journal of Economics, 107, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

30 Single-Threshold Model Heterogeneity is a common problem of panel data. That is to say, each individual in a study is different, and structural relationships may vary across individuals. The classical fixed effect or random effect reflects only the heterogeneity in intercepts. Hsiao (2003) considers many varying slope models for this problem. Among these models, Hansen s (1999) panel threshold model has a simple specification but obvious implications for economic policy. 7 Another is the panel smooth transition model of González, Teräsvirta, and van Dijk (2005). 8 Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

31 Single-Threshold Model Let s start with the single-threshold model: y it = µ i + β 1x it (q it < γ) + β 2x it (q it γ) + e it (13) where q it is the threshold variable, and γ is the threshold parameter that divides the equation into two regimes with coefficients β 1 and β 2. The parameter µ i is the individual (fixed) effect, while e it (0, σ 2 ) is the disturbance. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

32 Single-Threshold Model We can also write (13) as: where β = (β 1, β 2 ), and y it = µ i + β x it (γ) + e it (14) x it (γ) = and I( ) is an indicator function. ( xit I(q it < γ) x it I(q it γ) ) 7 Hansen, B. E. (1999), Threshold effects in non-dynamic panels: Estimation, testing, and inference. Journal of Econometrics, 93, González, A., Teräsvirta, T., and van Dijk, D. (2005), Panel smooth transition regression models. Research Paper 165, Quantitative Finance Research Centre, University of Technology, Sidney. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

33 Single-Threshold Model Estimation I By taking average of (14) over the time index t produces: ȳ i = µ i + β x i (γ) + ē i (15) where ȳ i = 1 T T t=1 y it; and similar notations apply to other variables. Taking the difference between (14) and (15) yields: yit = β x it(γ) + e it (16) where yit = y it ȳ i ; and other variables defined accordingly. By stacking all observations, we have: y = X (γ)β + e (17) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

34 Single-Threshold Model Estimation II Given γ, the ordinary least-squares estimator of β is: ˆβ(γ) = ( X (γ) X (γ) ) 1 X (γ) y (18) The vector of residuals is ê (γ) = y X (γ) ˆβ(γ), and the sum of squared errors is: S 1 (γ) = ê (γ) ê (γ) (19) Hence, the least squares estimator of γ is: ˆγ = arg min γ S 1 (γ) (20) Once ˆγ is obtained, the slope coefficient estimate is: ˆβ = ˆβ(ˆγ) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

35 Single-Threshold Model Estimation III The residual vector is ê = ê (ˆγ) and residual variance is: ˆσ 2 = 1 nt ê ê = 1 nt S 1(ˆγ) (21) The computation of the least squares estimate of the threshold γ involves the minimization problem (20). Since the sum of squared error function S1 (γ) depends on γ only through the indicator functions I(q it < γ), the sum of squared error function is a step function with at most N = nt steps, with the steps occurring at distinct values of the observed threshold variable q it. Thus the minimization problem (20) can be reduced to searching over values of γ equalling the (at most N) distinct values of q it in the sample. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

36 Single-Threshold Model Estimation IV To implement the minimization, the following approach may be taken. Sort the distinct values of the observations on the threshold variable qit. Eliminate the smallest and largest η% (say, 1% or 5%) for some η > 0. The remaining N values constitute the values of γ which can be searched for ˆγ. For each of these N values, regressions (18) are estimated yielding the sum of squared errors (19). The minimum value of the latter yields the estimate ˆγ. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

37 Single-Threshold Model Estimation V In practice, N may be a very large number, and the grid search described above may be numerically intensive. A simplifying shortcut which yields nearly identical results is to restrict the search to a smaller set of values of γ. Instead of searching over all values of qit (between the η% and (1 η)% quantile), the search may be limited to specific quantiles, perhaps integer valued. This greatly reduces the number of regressions performed in the search. In the later empirical application, we consider the following grids {1.00%, 1.25%, 1.50%,, 98.5%, 98.75%, 99.0%} which contain 393 quantiles. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

38 Single-Threshold Model Inference I It is important to determine whether the threshold effect is statistically significant. The hypothesis is: H 0 : β 1 = β 2 H 1 : β 1 β 2 Under the null hypothesis of no threshold, the model is: y it = µ i + β 1x it + e it (22) After the fixed-effect transformation is made, we have: y it = β 1x it + e it (23) The parameter β 1 is estimated by OLS, yielding estimate β 1, residuals ẽ it, and sum of squared errors S 0 = ẽ ẽ. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

39 Single-Threshold Model Inference II The likelihood ratio test of H 0 is based on: F 1 = S 0 S 1 (ˆγ) ˆσ 2 = S 0 S 1 (ˆγ) S 1 (ˆγ)/nT (24) which has a nonstandard asymptotic distribution. Under H0, the threshold γ is not identified, and the problem of nuisance parameter arises. Standard inferences are invalid, and we rely on the boostrapping procedure. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

40 Single-Threshold Model Inference III The bootstrap procedure: Step 1 Fit the model under H 1 and obtain the residual ê it. Step 2 Draw ê it with replacement, and obtain the new residual v it. Step 3 Generate a new series under the H 1 data-generating process, yit = β x it + v it, where β can take arbitrary values. Step 4 Fit the model under H 0 and H 1, and compute the F statistic using (24). Step 5 Repeat steps 1 4 for B times, and the p-value of F is the proportion of F > F 1 in bootstrap number B. The null of no threshold effect is rejected if the p-value is smaller than the desired significance value. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

41 Single-Threshold Model Panel Smooth Transition Regression Model I González et al. (2005) propose the following PSTR model: 9 y it = µ i + β 1x it + β 2x it g(q it ; γ, c) + e it (25) The model can be interpreted in two different ways. First, it may be thought of as a linear heterogenous panel model with coefficients that vary across individuals and over time. Second, the PSTR model can simply be considered as a nonlinear homogenous panel model. Transition function g(qit ; γ, c) is a continuous function of the observable variable q it and is normalized to be bounded between 0 and 1, and these extreme values are associated with coefficients β 1 and β 1 + β 2. More generally, the value of q it determines the value of g(q it; γ, c) and thus the effective coefficients β 1 + β 2g(q it; γ, c) for individual i at time t. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

42 Single-Threshold Model Panel Smooth Transition Regression Model II The transition function can take the (logistic) form: g(q it ; γ, c) = exp( γ(q it c)) (26) where the slope parameter γ > 0 (determines the smoothness of the transitions), and c is the location parameter. The model implies that the two extreme regimes are associated with low and high values of q it with a single monotonic transition of the coefficients from β 1 and β 1 + β 2 as q it increases, where the change is centred around c. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

43 Single-Threshold Model Panel Smooth Transition Regression Model III When γ, g(qit ; γ, c) becomes an indicator function I[q it > c], defined as I[A] = 1 when the event A occurs and 0 otherwise. In that case the PSTR model in (25) reduces to the two-regime panel threshold model of Hansen (1999). The transition function (26) becomes constant when γ 0, in which case the model collapses into a homogenous or linear panel regression model with fixed effects. 9 Shen, C. H. (2005), Cost efficiency and banking performances in a partial universal banking system: Application of the panel smooth threshold model. Applied Economics, 37, Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

44 Multiple-Threshold Model If there are multiple thresholds (that is, multiple regimes), we can fit the model sequentially. We use a double-threshold (hence, three regimes) model as an example: y it = µ i + β 1x it (q it < γ 1 ) + β 2x it (γ 1 q it < γ 2 ) +β 3x it (q it γ 2 ) + e it (27) where the thresholds are ordered so that γ 1 < γ 2. Here, γ1 and γ 2 are the thresholds that divide the equation into three regimes with coefficients β 1, β 2, and β 3, respectively. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

45 Multiple-Threshold Model Estimation I For given (γ 1, γ 2 ), (27) is linear in the slopes (β 1, β 2, β 3 ) so OLS estimation is appropriate. Thus for given (γ1, γ 2 ) the concentrated sum of squared errors S(γ 1, γ 2 ) is straightforward to calculate. The joint LS estimates of (γ 1, γ 2 ) are by definition the values which jointly minimize S(γ 1, γ 2 ). While these estimates might seem desirable, they may be quite cumbersome to implement in practice. A grid search over (γ1, γ 2 ) requires approximately N 2 = (nt ) 2 regressions which may be prohibitively expensive. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

46 Multiple-Threshold Model Estimation II However, it has been found (Bai, 1997; Bai and Perron, 1998) in the multiple changepoint model that sequential estimation is consistent. The same logic appears to apply to the multiple threshold model. Step 1 In the first stage, let S 1 (γ) be the single threshold sum of squared errors as defined in (19) and let ˆγ 1 be the threshold estimate which minimizes S 1 (γ). Step 2 Fixing the first-stage estimate ˆγ 1, the second-stage criterion is: { S2(γ r S(ˆγ1, γ 2 ) = 2 ) if ˆγ 1 < γ 2 (28) S(γ 2, ˆγ 1 ) if γ 2 < ˆγ 1 and the second-stage threshold estimate is: ˆγ r 2 = arg min γ 2 S r 2(γ 2 ) (29) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

47 Multiple-Threshold Model Estimation III Step 3 Bai (1997) has shown that ˆγ r 2 is asymptotically efficient, but ˆγ 1 is not. This is because the estimate ˆγ 1 was obtained from a sum of squared errors function which was contaminated by the presence of a neglected regime. The asymptotic efficiency of ˆγ r 2 suggests that ˆγ 1 can be improved by a third-stage estimation. Fixing the second-stage estimate ˆγ r 2, define the refinement criterion as: and the refinement estimate is: S r 1(γ 1 ) = { S(γ1, ˆγ r 2) if γ 1 < γ r 2 S(ˆγ r 2, γ 1 ) if ˆγ r 2 < γ 1 (30) ˆγ r 1 = arg min γ 1 S r 1(γ 1 ) (31) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

48 Multiple-Threshold Model Inference The threshold-effect test is also sequential; that is, if we reject the null hypothesis in a single-threshold (two regimes) model, then we must test the double-threshold (three regimes) model. The null hypothesis is a single-threshold model, and the alternative hypothesis is a double-threshold model. The F statistic is constructed as: F 2 = S 1(ˆγ 1 ) S r 2(ˆγ r 2) ˆσ 2 where S r 2(ˆγ r 2) is the minimizing sum of squared errors from the second-stage threshold estimate, and the variance estimate is ˆσ 2 = 1 nt Sr 2(ˆγ r 2). Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

49 Investment and Financing Constraints Application I To explore the link between investment and financing constraints: Classical models of the firms assume the existence of perfect financial markets on which firms can borrow the needed resources for investment projects. Alternative models of financing place restrictions on the extent of external financing. Fazzari et al. (1988) argue that the presence of financing constraints implies that a firm s cash flow will be positively related to its investment rate only when the firm faces constraints on external financing. If a firm is free to borrow on external financial markets, cash flow will be irrelevant for investment. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

50 Investment and Financing Constraints Application II We use the multiple/double threshold regression model: where I it = µ i + θ 1 Q i,t 1 + θ 2 Q 2 i,t 1 + θ 3 Q 3 i,t 1 +θ 4 D i,t 1 + θ 5 Q i,t 1 D i,t 1 +β 1 CF i,t 1 I(D i,t 1 γ 1 ) +β 2 CF i,t 1 I(γ 1 < D i,t 1 γ 2 ) +β 3 CF i,t 1 I(γ 2 < D i,t 1 ) + e it (32) Iit is the ratio of investment to capital. Q it is the ratio of total market value to assets. Dit is the ratio of long-term debt to assets. CFit is the ratio of cash flow to assets. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

51 Investment and Financing Constraints Application III Equation (32) is a double threshold model, and falls in the class of model (27) setting q it = D i,t 1 and x it = CF i,t 1. In addition to x it, there are also the additional regressors (Q i,t 1, Q 2 i,t 1, Q3 i,t 1, D i,t 1, Q i,t 1 D i,t 1 ). Since the method is designed for balanced panels, so Hansen (1999) took the subset of 565 firms (from the unbalanced panel of Hall and Hall (1993)), which are observed for the years The Stata command xtbalance can be used to construct a balanced panel. (Demonstrate if time permits) 10 ssc install xtbalance, replace Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

52 Investment and Financing Constraints Application IV Code/data: Stata: Please first install relevant softwares: 11 ssc install xthreg, replace and ssc install outreg2, replace Code/data: thrpd-hansen09je.do/dta. 10 Please see illustrative file: xtbalance.do. Very useful! 11 Wang, Q. (2015), Fixed-effect panel threshold model using Stata. Stata Journal, 15(1), Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

53 Linear Model Tuesday May 24 15:51:29 Threshold 2016 ModelPage 1 Panel Threshold Model Applications Investment and Financing Constraints xthreg I Statistics/Data Analysis help xthreg (SJ15 1: st0373) Title xthreg Estimate fixed effect panel threshold model Syntax xthreg depvar [indepvars] [if] [in], rx(varlist) qx(varname) [thnum(#) grid(#) trim(numlist) bs(numlist) thlevel(#) gen(newvarname) noreg nobslog thgiven options] where depvar is the dependent variable and indepvars are the regime independent variables. Description xthreg fits fixed effect panel threshold models based on the method proposed by Hansen (1999). xthreg uses [XT] xtreg to fit the fixed effect panel threshold model given the threshold estimator. The fixed effect panel threshold model requires balanced panel data, which is checked automatically by xthreg. The estimation and test of the threshold effect are computed in Mata. Options Ho-Chuan rx(varlist) (River) Huang is the regime dependent 面板門檻模型 variable. Time series operators Juneare 5 7, / 62

54 Ho-Chuan thlevel(#) (River) Huang specifies the confidence 面板門檻模型 level, as a percentage, for confidence June 5 7, / 62 Linear Model Threshold Model Panel Threshold Model Applications Description Investment and Financing Constraints xthreg II xthreg fits fixed effect panel threshold models based on the method proposed by Hansen (1999). xthreg uses [XT] xtreg to fit the fixed effect panel threshold model given the threshold estimator. The fixed effect panel threshold model requires balanced panel data, which is checked automatically by xthreg. The estimation and test of the threshold effect are computed in Mata. Options rx(varlist) is the regime dependent variable. Time series operators are allowed. rx() is required. qx(varname) is the threshold variable. Time series operators are allowed. qx() is required. thnum(#) is the number of thresholds. In the current version (Stata 13), # must be equal to or less than 3. The default is thnum(1). grid(#) is the number of grid points. grid() is used to avoid consuming too much time when computing large samples. The default is grid(300). trim(numlist) is the trimming proportion to estimate each threshold. The number of trimming proportions must be equal to the number of thresholds specified in thnum(). The default is trim(0.01) for all thresholds. For example, to fit a triple threshold model, you may set trim( ). bs(numlist) is the number of bootstrap replications. If bs() is not set, xthreg does not use bootstrap for the threshold effect test.

55 Investment and Financing Constraints xthreg III be equal to or less than 3. The default is thnum(1). grid(#) is the number of grid points. grid() is used to avoid consuming too much time when computing large samples. The default is grid(300). trim(numlist) is the trimming proportion to estimate each threshold. The number of trimming proportions must be equal to the number of thresholds specified in thnum(). The default is trim(0.01) for all thresholds. For example, to fit a triple threshold model, you may set trim( ). bs(numlist) is the number of bootstrap replications. If bs() is not set, xthreg does not use bootstrap for the threshold effect test. thlevel(#) specifies the confidence level, as a percentage, for confidence intervals of the threshold. The default is thlevel(95). gen(newvarname) generates a new categorical variable with 0, 1, 2,... for each regime. The default is gen(_cat). noreg suppresses the display of the regression result. nobslog suppresses the iteration process of the bootstrap. thgiven fits the model based on previous results. options are any options available for [XT] xtreg. Time series operators are allowed in depvar, indepvars, rx(), and qx(). Examples Setup Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

56 Investment and Financing Constraints Results. tabstat i q1 cf1 d1, stat(min p25 p50 p75 max) col(stat) variable min p25 p50 p75 max i q cf d Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

57 Investment and Financing Constraints. xtreg i q1 q2 q3 d1 qd1 cf1, fe Fixed-effects (within) regression Number of obs = 7,910 Group variable: id Number of groups = 565 R-sq: Obs per group: within = min = 14 between = avg = 14.0 overall = max = 14 F(6,7339) = corr(u_i, Xb) = Prob > F = i Coef. Std. Err. t P> t [95% Conf. Interval] q q q d qd cf _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(564, 7339) = 7.36 Prob > F = Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

58 Investment and Financing Constraints. xthreg i q1 q2 q3 d1 qd1, rx(cf1) qx(d1) thnum(3) grid(400) /// > trim( ) bs( ) noreg Threshold estimator (level = 95): model Threshold Lower Upper Th Th Th Th Threshold effect test (bootstrap = ): Threshold RSS MSE Fstat Prob Crit10 Crit5 Crit1 Single Double Triple Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

59 Investment and Financing Constraints Application. xthreg i q1 q2 q3 d1 qd1, rx(cf1) qx(d1) thnum(2) grid(400) /// > trim( ) vce(robust) Threshold estimator (level = 95): model Threshold Lower Upper Th Th Th Fixed-effects (within) regression Number of obs = 7910 Group variable: id Number of groups = 565 R-sq: within = Obs per group: min = 14 between = avg = 14.0 overall = max = 14 F(8,564) = Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

60 Th Linear Model Th Threshold Model Panel Threshold Model Applications Investment and Financing Constraints Fixed-effects (within) regression Number of obs = 7910 Group variable: id Number of groups = 565 R-sq: within = Obs per group: min = 14 between = avg = 14.0 overall = max = 14 F(8,564) = corr(u_i, Xb) = Prob > F = (Std. Err. adjusted for 565 clusters in id) Robust i Coef. Std. Err. t P> t [95% Conf. Interval] q q q d qd _cat#c.cf _cons sigma_u sigma_e rho (fraction of variance due to u_i) Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

61 Inequality on Growth Practice I This exercise explores whether the effect of income inequality on economic growth (whether positive or negative) depends on the level/stage of economic development. The data were kindly provided by Mark W. Frank (2009), and used in his paper entitled Inequality and growth in the United States: Evidence from a new state-level panel of income inequality measures. Economic Inquiry, 47, He finds that the long-run relationship (by the PMG approach) between inequality and growth is positive in nature and driven principally by the concentration of income in the upper end of the income distribution. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62

62 Inequality on Growth Practice II Consider the (multiple) panel threshold model: g it = µ i + α w it + J β j ineq it I(γ j 1 < q it γ j ) + e it (33) j=1 where g it is the growth rate of real per capita state income in state i at time t, µ i is the state-specific effect, w it is a set of growth determinants, ineq it = ln(top10 it ) is a measure of income inequality, and q it (threshold variable) is the (lagged) logarithm of state real income per capita. Code/data: thrpd-frank09ei.do/dta. Ho-Chuan (River) Huang 面板門檻模型 June 5 7, / 62