1 Introduction
R
functions which are available on Github at the web page https://github.com/Silvia-Pand/BivLT.2 The statistical model
2.1 Basic notation
2.2 Model assumptions
3 Model inference
3.1 Maximum likelihood estimation
-
E-step: it consists in computing the conditional expected value of the complete data log-likelihood given the observed data and the current value of the model parameters.
-
M-step: it consists in maximizing the expected value of the complete data log-likelihood resulting from the E-step with respect to \( {\varvec{\theta }}\), so as to update the parameters.
3.2 Standard errors, model selection, and clustering
3.3 Marginal effects
4 Application
R
code to specify the bivariate latent growth model and display the model parameter estimates.4.1 Data description
1998 | 2000 | 2002 | 2004 | 2006 | 2008 | 2010 | 2012 | 2014 | |
---|---|---|---|---|---|---|---|---|---|
Risky asset market participation | 30.32 | 35.49 | 31.80 | 30.93 | 28.76 | 24.77 | 32.81 | 32.64 | 29.78 |
Conditional investment share | 45.20 | 43.32 | 41.71 | 43.74 | 41.35 | 37.26 | 48.16 | 46.99 | 43.77 |
4.2 Model specification
disposable income
, in thousands of euros), whether household has any debt (dummy debts
), number of household members (household size
), presence of children under 14 years (dummy children
), marital status (dummy married
), and employment status of the household head (dummies employee
and retired
). As common practice in estimating selection models, in order to improve model identifiability we impose an exclusion restriction and assume that asset market participation probability is also affected by the stock of real assets (real assets
, in thousands of euros) owned by the household, by the regional unemployment rate (unemployment rate
), and by the average number of bank branches (per 100,000 inhabitants) at regional level (bank branch density
).female
) and the values observed at the first available time occasion for the area of residence (dummies north
and centre
), town size (dummy small town
), and household head’s educational level (dummies lower secondary education
, upper secondary education
, and tertiary education
).youth stock return
in the following), defined as a weighted average of the Italian Stock Exchange (80%) and the MSCI World Index (20%), experienced when the head was aged between 18 and 25. As this composite indicator is time-constant, we include it in vector \({\mathbf {z}}_i\). However, it is worth noting that its inclusion in vectors \({\mathbf {w}}_{it}\) and \({\mathbf {x}}_{it}\) does not modify in a sensible way the results of the estimation process (results not shown here).4.3 Model selection and latent class characterization
Model | Polynomial for age | Polynomial for year | \({\hat{\ell }}\) | BIC | \({\hat{\rho }}\) | s.e. \({\hat{\rho }}\) | \({\hat{\sigma }}^2\) | s.e. \({\hat{\sigma }}^2\) |
---|---|---|---|---|---|---|---|---|
Heckman | \(r=3\) | \(r=3\) | \(-\) 10,010.99 | 20,287.87 | \(-\) 0.239 | 0.068 | 0.093 | 0.004 |
Heckman | \(\mathbf {r=4}\) | \(\mathbf {r=4}\) | \(-\) 9984.13 | 20,266.37 | \(-\) 0.244 | 0.068 | 0.093 | 0.004 |
Heckman | \(r=5\) | \(r=5\) | \(-\) 9975.44 | 20,281.23 | \(-\) 0.250 | 0.068 | 0.093 | 0.004 |
Latent growth (\(k=1\)) | \(r=3\) | \(r=3\) | \(-\) 10,011.32 | 20,288.53 | \(-\) 0.241 | 0.047 | 0.093 | 0.002 |
Latent growth (\(k=1\)) | \(\mathbf {r=4}\) | \(\mathbf {r=4}\) | \(-\) -9984.47 | 20,267.06 | \(-\) 0.246 | 0.047 | 0.093 | 0.002 |
Latent growth (\(k=1\)) | \(r=5\) | \(r=5\) | \(-\) 9975.80 | 20,281.94 | \(-\) 0.252 | 0.046 | 0.093 | 0.002 |
Latent growth (\(k=1\)) | \(r=3\) | Dummies | \(-\) 9972.04 | 20,290.55 | \(-\) 0.252 | 0.047 | 0.093 | 0.002 |
Latent growth (\(k=1\)) | \(\mathbf {r=4}\) | Dummies | \(-\) 9955.27 | 20,273.11 | \(-\) 0.251 | 0.046 | 0.093 | 0.002 |
Latent growth (\(k=1\)) | \(r=5\) | Dummies | \(-\) 9954.73 | 20,288.15 | \(-\) 0.251 | 0.046 | 0.093 | 0.002 |
k | \({\hat{\ell }}\) | BIC | \({\hat{\rho }}\) | s.e. (\({\hat{\rho }}\)) | BIC (\(\rho =0\)) |
---|---|---|---|---|---|
1 | \(-\) 9984.47 | 20,267.06 | \(-\) 0.246 | 0.047 | 20,295.50 |
2 | \(-\) 8648.46 | 17,965.68 | \(-\) 0.220 | 0.063 | 17,987.27 |
3 | \(-\) 8290.64 | 17,620.68 | \(-\) 0.115 | 0.090 | 17,636.96 |
4 | \(-\) 8129.68 | 17,669.40 | \(-\) 0.070 | 0.104 | 17,683.36 |
Class 1 | Class 2 | Class 3 | |
---|---|---|---|
Class size | 517 | 1709 | 931 |
Proportion | 0.164 | 0.541 | 0.295 |
Time-varying covariates | |||
disposable income (\(\times 1000\)) | 46.061 | 24.957 | 37.069 |
children | 0.171 | 0.161 | 0.218 |
married | 0.741 | 0.613 | 0.710 |
household size (number) | 2.569 | 2.515 | 2.680 |
employee | 0.534 | 0.356 | 0.560 |
retired | 0.432 | 0.540 | 0.391 |
debts | 0.215 | 0.164 | 0.224 |
real assets (\(\times 1000\)) | 391.660 | 163.360 | 264.309 |
unemployment rate (%) | 6.383 | 10.553 | 7.462 |
bank branch density | 6.540 | 5.055 | 6.134 |
Time-constant covariates | |||
north | 0.710 | 0.335 | 0.604 |
centre | 0.180 | 0.177 | 0.215 |
small town | 0.251 | 0.300 | 0.328 |
female | 0.201 | 0.401 | 0.282 |
lower secondary education | 0.271 | 0.348 | 0.354 |
upper secondary education | 0.410 | 0.153 | 0.371 |
tertiary education | 0.259 | 0.054 | 0.125 |
youth stock return (%) | 8.804 | 10.577 | 11.169 |
4.4 Life-cycle and time patterns of households’ investment behavior
(1) Participation probability | (2) Share invested | |||||
---|---|---|---|---|---|---|
\(\hat{{\varvec{\beta }}}_{1}\) | \(\hat{{\varvec{\beta }}}_{2}\) | \(\hat{{\varvec{\beta }}}_{3}\) | \(\hat{{\varvec{\gamma }}}_{1}\) | \(\hat{{\varvec{\gamma }}}_{2}\) | \(\hat{{\varvec{\gamma }}}_{3}\) | |
age | 0.062 | \(-\) 0.059 | \(-\) 0.132\(^{(.)}\) | 0.036* | \(-\) 0.065 | 0.016 |
(0.093) | (0.072) | (0.070) | (0.015) | (0.041) | (0.019) | |
age \(^\mathtt{2}\) | 0.050 | \(-\) 0.243*** | \(-\) 0.039 | 0.002 | 0.094** | 0.008 |
(0.058) | (0.060) | (0.039) | (0.010) | (0.033) | (0.012) | |
age \(^\mathtt{3}\) | \(-\) 0.006 | 0.023 | 0.004 | \(-\) 0.001 | 0.005 | 0.002 |
(0.017) | (0.015) | (0.015) | (0.003) | (0.008) | (0.005) | |
age \(^\mathtt{4}\) | \(-\) 0.009 | 0.030*** | \(-\) 0.007 | \(-\) 0.002 | \(-\) 0.012* | 0.000 |
(0.009) | (0.008) | (0.007) | (0.002) | (0.005) | (0.002) | |
year | \(-\) 0.044 | \(-\) 0.006 | \(-\) 0.148*** | \(-\) 0.011\(^{(.)}\) | 0.097*** | \(-\) 0.039*** |
(0.038) | (0.035) | (0.025) | (0.006) | (0.022) | (0.006) | |
year \(^\mathtt{2}\) | 0.341*** | 0.720*** | 1.029*** | 0.069\(^{(.)}\) | 0.119 | \(-\) 0.039 |
(0.232) | (0.221) | (0.144) | (0.037) | (0.112) | (0.044) | |
year \(^\mathtt{3}\) | 0.223 | 0.098*** | 0.704 | 0.065 | \(-\) 0.503** | 0.116* |
(0.274) | (0.271) | (0.176) | (0.048) | (0.168) | (0.051) | |
year \(^\mathtt{4}\) | \(-\) 1.373 | \(-\) 2.374* | \(-\) 5.101*** | \(-\) 0.540* | \(-\) 0.711 | 0.271 |
(1.333) | (1.251) | (0.842) | (0.226) | (0.680) | (0.258) |
4.5 Effect of time-varying and time-constant covariates
(1) Participation probability | (2) Share invested | |||||
---|---|---|---|---|---|---|
\(\hat{{\varvec{\beta }}}_{1}\) | \(\hat{{\varvec{\beta }}}_{2}\) | \(\hat{{\varvec{\beta }}}_{3}\) | \(\hat{{\varvec{\gamma }}}_{1}\) | \(\hat{{\varvec{\gamma }}}_{2}\) | \(\hat{{\varvec{\gamma }}}_{3}\) | |
disposable income | 0.088*** | 0.098*** | 0.233*** | 0.003 | \(-\) 0.014** | 0.000 |
(0.024) | (0.015) | (0.022) | (0.002) | (0.005) | (0.003) | |
children | 0.187 | 0.064 | \(-\) 0.082 | \(-\) 0.002 | \(-\) 0.118* | \(-\) 0.002 |
(0.135) | (0.107) | (0.076) | (0.022) | (0.057) | (0.021) | |
married | \(-\) 0.028 | 0.014 | 0.228*** | 0.053** | \(-\) 0.087\(^{(.)}\) | 0.018 |
(0.113) | (0.092) | (0.071) | (0.019) | (0.050) | (0.021) | |
household size | 0.041 | 0.102** | \(-\) 0.136*** | \(-\) 0.023** | \(-\) 0.032\(^{(.)}\) | -0.008 |
(0.054) | (0.039) | (0.032) | (0.008) | (0.018) | (0.008) | |
employee | \(-\) 0.198 | 0.226 | 0.112 | 0.038 | \(-\) 0.256*** | \(-\) 0.024 |
(0.182) | (0.145) | (0.115) | (0.034) | (0.078) | (0.036) | |
retired | \(-\) 0.161 | 0.127 | 0.041 | \(-\) 0.002 | \(-\) 0.284*** | 0.083* |
(0.207) | (0.152) | (0.131) | (0.037) | (0.085) | (0.039) | |
debts | \(-\) 0.123 | 0.074 | \(-\) 0.030 | 0.016 | 0.077* | \(-\) 0.024 |
(0.088) | (0.083) | (0.058) | (0.016) | (0.040) | (0.016) | |
real assets | 0.001 | 0.003*** | 0.004*** | |||
(0.001) | (0.001) | (0.001) | ||||
unemployment rate | \(-\) 0.033\(^{(.)}\) | \(-\) 0.038* | 0.028* | |||
(0.020) | (0.016) | (0.012) | ||||
bank branch density | 0.133* | 0.046 | 0.178*** | |||
(0.055) | (0.038) | (0.032) | ||||
constant | \(-\) 0.165 | \(-\) 2.412*** | \(-\) 2.429*** | 0.679*** | 0.962*** | 0.399*** |
(0.497) | (0.362) | (0.305) | (0.041) | (0.114) | (0.053) |
(1) Participation probability | (2) Share invested | |
---|---|---|
disposable income | 0.003*** | \(-\) 0.006 |
(0.000) | (0.006) | |
children | 0.005 | \(-\) 0.060*** |
(0.014) | (0.006) | |
married | 0.023\(^{(.)}\) | \(-\) 0.028 |
(0.015) | (0.033) | |
household size | \(-\) 0.006 | \(-\) 0.023 |
(0.006) | (0.015) | |
employee | 0.016 | \(-\) 0.129* |
(0.021) | (0.060) | |
retired | 0.004 | \(-\) 0.114\(^{(.)}\) |
(0.022) | (0.065) | |
debts | \(-\) 0.005 | 0.033 |
(0.010) | (0.031) | |
real assets | 0.001** | |
(0.000) | ||
unemployment rate | \(-\) 0.001 | |
(0.003) | ||
bank branch density | 0.028*** | |
(0.006) |
(1) Class 2 membership | (2) Class 3 membership | |
---|---|---|
\(\hat{{\varvec{\delta }}}_{2}\) | \(\hat{{\varvec{\delta }}}_{3}\) | |
north | \(-\) 2.420*** | \(-\) 0.905*** |
(0.239) | (0.246) | |
centre | \(-\) 1.609*** | \(-\) 0.514\(^{(.)}\) |
(0.256) | (0.272) | |
small town | 0.085 | 0.203 |
(0.154) | (0.155) | |
female | 0.924*** | 0.556*** |
(0.170) | (0.171) | |
lower secondary education | \(-\) 1.596*** | \(-\) 0.652* |
(0.253) | (0.293) | |
upper secondary education | \(-\) 2.971*** | \(-\) 1.198*** |
(0.269) | (0.296) | |
tertiary education | \(-\) 3.534*** | \(-\) 1.810*** |
(0.297) | (0.322) | |
youth stock return | \(-\) 0.003 | 0.013* |
(0.007) | (0.007) | |
constant | 4.345*** | 2.018*** |
(0.333) | (0.382) |
(1) Participation probability | (2) Share invested | |
---|---|---|
north | 0.167*** | \(-\) 0.023 |
(0.019) | (0.041) | |
centre | 0.107*** | \(-\) 0.013 |
(0.016) | (0.030) | |
small town | \(-\) 0.002 | \(-\) 0.007 |
(0.008) | (0.008) | |
female | \(-\) 0.056*** | \(-\) 0.005 |
(0.010) | (0.011) | |
lower secondary education | 0.102*** | \(-\) 0.009 |
(0.010) | (0.023) | |
upper secondary education | 0.202*** | \(-\) 0.012 |
(0.014) | (0.040) | |
tertiary education | 0.250*** | 0.024*** |
(0.020) | (0.004) | |
youth stock return | 0.000 | \(-\) 0.001\(^{(.)}\) |
(0.000) | (0.000) |