4. A hidden Markov model for panel data 64
4.5. Switching Regression: Cross sectional analysis with covariables
0.00.20.40.60.81.0
time
p−values
1970 1980 1990 2000 2010
0.00.20.40.60.8
time
p−values
1970 1980 1990 2000 2010
0.000.050.100.150.20
time
p−values
1970 1980 1990 2000 2010
Figure 4.13.: Reduced model: p-values of regression for mixing probabilities. Left:
income group 1, mid: income group 2, right: income group 3. Covariables intercept (solid line), years of schooling (dashed line), life expectancy (dotted line), latitude
(dash-dotted line). Longdash: 5% and 10% level.
variables intercept,years of schooling,latitude and life expectancy.
The p-values of the reduced models are shown in Figure 4.13. We observe that the vari-able life expectancy is highly significant in the three income groups with p-value close to zero. In the first income group beginning in 1990 the p-value of the variable latitude rises over the 10% significance level, while the remaining covariables are significant at the 5% level over time (except of the intercept in a period between 1990 and 1992). In the second income group the p-value of the intercept is quite high from 1970–1992 and while close to zero in the beginning of the observed time horizon, from 1990 the p-value of the variable years of schooling rises and stays over the 10% level after 1999. In the third income group all variables have p-values close to zero over the 41 years, except of the variable latitude with p-value close to zero from 1970 until 1993, rising above the 10% significance level between 1997 and 2000 and after 2007.
The estimated coefficients of the reduced models show that the chosen covariablesyears of schooling, life expectancy and latitude seem to have positive effects on the GDP of a country. We observe that in income group 1 the signs of the estimated coefficients are almost always negative. Thus, increasing years of schooling, life expectancy and latitude lowers the probability of a country to be in income group 1. In income group 2 the variables years of schooling and latitude still have negative signs, while the variable life expectancy has positive influence on the probability of a country to be part of income group 2. In income group 3 the effects of all three covariables have positive signs.
4.5. Switching Regression: Cross sectional analysis with
intercept) and write χ(t)i = (χ(t)1,i, . . . , χ(t)L,i)T for the data of country i ∈ {1, . . . , I} at timet∈ {1, . . . , T}.
For each year we formulate aK-component Gaussian mixture model f(xt,i) =
K
∑
k=1
πk,i(t)g(xt,i;ϑ(t)k ),
with mixing probabilities modelled by categorial logit regression: Letr denote the ref-erence income group of the model and M ={1, . . . , K} \ {r}. In this model, we choose r= 2, thus the results are to be interpreted relative to income group 2. Then, fork∈M,
π(t)k,i= exp(χ(t)Ti βk(t)) 1 +∑
l∈Mexp(χ(t)l βl(t))
, πr,i(t)= 1−∑
l∈M
πl,i(t), (i= 1, . . . , I, t= 1, . . . , T), (4.1) thusβk(t)= (βk,1(t), . . . , βk,L(t)) (k∈M) denote the parameter vectors of the regression part of the model.
For estimation of the (K−1)T L+ 2KT parametersβk(t)(k∈M) andϑ(t)k (k= 1, . . . , K, t= 1, . . . , T), we maximize the likelihood function of the model, given by
LSRT =
T
∏
t=1 I
∏
i=1
[∑
k∈M
exp(χ(t)Ti βk(t)) 1 +∑
l∈Mexp(χ(t)Ti βl(t))
g(xt,i;ϑ(t)k )
+ (1−∑
l∈M
exp(χ(t)Ti βl(t)) 1 +∑
m∈Mexp(χ(t)Ti βm(t))
)g(xt,i;ϑ(t)r )].
3.03.54.04.5
time
mean
1970 1980 1990 2000 2010
0.100.150.200.25
time
standard deviation
1970 1980 1990 2000 2010
Figure 4.14.:Parameter estimates switching regression - left: means, right: standard deviations. Income group 1 (solid line), income group 2 (dashed line), income group 3
(dotted line).
Maximization is performed using an EM-algorithm where the group membership is the latent variable and due to the independence assumption, each year can be modelled separately. Details on the computation are given in Section 4.8.2.
The estimated parameters of the component-dependent distributions are shown in Figure 4.14. We observe that the estimated means are less volatile than in the mixture model, but of the same magnitude. In addition, the rising mean in the 1990s followed by a decline after 1996 combined with a high standard deviation reminds of the results in the mixture model and the modified hidden Markov models presented in the previous sections.
−80−60−40−200
time
coefficients
1970 1980 1990 2000 2010
(a) Intercept
−1001020
time
coefficients
1970 1980 1990 2000 2010
(b)Years of schooling
−20020406080
time
coefficients
1970 1980 1990 2000 2010
(c) Life expectancy
0246810
time
coefficients
1970 1980 1990 2000 2010
(d) Latitude
Figure 4.15.:Switching Regression: Estimated parameters in multinomial regression.
Income group 1 (solid line), income group 3 (dashed line).
The estimated parameters of the multinomial regression are shown in Figure 4.15. Since income group 2 is the reference category of our model, the estimates can be interpreted relative to this income group. The estimates for variable years of schooling in income group 1 are negative, thus the odds for income group 1 relative to income group 2 decrease with increasing years of schooling, while the odds for income group 3 relative to income group 2 increase, due to the positive sign of the estimated parameters. The same effect holds for the variable life expectancy. For the variable latitude we observe that for income group 1 the sign of the parameter estimate changes from negative to positive in 1990. Thus, from 1970 to 1990, increasing latitude decreases the odds for income group 1 relative to income group 2, while after 1990, the odds increase. The parameter for income group 3 is positive from 1970–2010.
Based on the estimation results we perform maximum-aposteriori analysis and report the results in Section 4.9. We observe that compared to the results from the mixture model, the number of switches of income group decreased dramatically. One reason is the number of countries for which we observe data, which decreased from 152 in models without covariables to I = 107 when using covariables. In the mixture model we observed switches of income groups for 76 countries, 24 of these can not be modelled in the switching regression model due to missing data. 35 countries which switched income group at least once in the mixture model are assigned to a fix income group over the 41 years in the switching regression model. This effect might be a result of the parameter estimates which are much more stable, as mentioned above. In addition to the decreasing number of countries which switch income group, the number of back-and-forth-switches as observed very often in the mixture model decreased, but the effect still occurs (see for example China (CHN), Iraq (IRQ), Morocco (MAR), Nicaragua
0.250.300.350.400.45
time
shares of income groups
1970 1980 1990 2000 2010
Figure 4.16.: Switching Regression: shares of income groups. Income group 1 (solid line), income group 2 (dashed line), income group 3 (dotted line).
(NIC), Portugal (PRT) and Vietnam (VNM)). Even though switching income group during the 41 years, 8 countries end up in the same group in 2010 as they started in 1970 and thus do not experience an advancement in the end. These are Republic of Congo (COG), Iraq (IRQ), Sri Lanka (LKA), Morocco (MAR), Mongolia (MNG), Nicaragua (NIC), Papua new guinea (PNG) and Swaziland (SWZ). It bears mentioning that the 9 countries which experience a switch of income group in the switching regression model all ascent. Namely these are Botswana (BWA), China (CHN), Cyprus (CYP), Egypt (EGY), Indonesia (IDN), Republic of Korea (KOR), Portugal (PRT), Thailand (THA) and Vietnam (VNM). Five of these countries were also modelled as one of the 24 ascending countries in the hidden Markov model (CHN, CYP, EGY, IDN and KOR).
The hidden Markov model which modelled ascending and declining countries separately also covered the advancement of PRT in addition to these five countries.
The shares of income group are shown in Figure 4.16. We observe that income group 2 is the largest income group all over the time, except for 1998, when income group 1 and income group 2 have the same share. The share of income group 1 decreases from 1970 until 1989 and then fluctuates around 1/3. Over the 41 years, the share of income group 3 slightly rises from 21.5% to 25.3%.