Seppo Suominen Essays on cultural economics

4.3Method and sample

The most recent data, ISSP 2007 (International Social Survey Programme) is based on a mail survey that was carried out by the Statistics Finland Autumn 2007 (18^th September – 11^th December 2007). The sample unit is a person, age between 15 and 74. The sample method was a systematic random sample from the population register, the sample size was 2500 but only 1354 answers were returned, i.e. the response rate was 54.2%. The key words in the ISSP 2007 are the following: use of time, physical condition, hobbies, organisations, board games, physical education, holiday, games, social relations, sports, leisure. As background information the following data among others was collected: gender, year of birth, size of the household, education, participation in working life, profession, source of livelihood or branch, regular weekly working hours, professional status, employer (private or public sector), membership of a trade union, voting behaviour, religiousness, income and some information concerning the place of residence.

The test statistics is distributed according to the F-distribution. If the difference between groups is significant, the difference can be evaluated with ή² = SS_between/SS_total which tells how much of the variation of the variable to be explained can be explained by the grouping variables (Metsämuuronen 2009, 785-789).

(4-8)

where X includes all explanatory variables and β is the vector of coefficients, u is the error term. The statistical significance of β can be evaluated with a suitable test. Usually it is assumed that the error term is distributed according to logistic (Weibull) distribution or to normal distribution. In the last case, the model is probit. Both logit and probit give more information compared with the analysis of variance because both the coefficients of the explanatory variables and the direction of the effect are found out: positive or negative and its statistical significance. Usually the hypothesis testing (single variable) is based on usual t-tests using the standard errors. A common test, which is similar to the F-test that all slope parameters in a regression are zero, is the likelihood ratio test that all the slope coefficients in the probit or logit model are zero.
The normal distribution for the binary choice (no = 0 / yes = 1) has been used frequently generating the probit model.

(4-9)

The function is the commonly used notation for the standard normal distribution (Greene 2008, 773) and x is a vector of explanatory variables and β is the corresponding vector of parameters. The logistic distribution which is mathematically convenient has been very popular.

(4-10)

(4-11)

y = 1, if 0 < y* ≤ µ₁

y = 2, if µ₁ < y* ≤ µ₂

y = 3, if µ₂ < y* ≤ µ₃

y = 4, if µ₃ ≤ y*

The µ’s are unknown parameters to be estimated with β. If ε is normally distributed with zero mean and variance equal to one [ε~N(0,1)] the following probabilities ensue (Greene 2008, 831-832):

(4-12)

The marginal effects in the multivariate probit²⁰ are

(4-13)

One step forward is to simultaneously study the visitor density of different leisure activities, e.g. “performing arts” on the one hand and “at the movies” or “physical exercise activity”. If the unobserved person’s preference for performing arts is y₁* and the preference for movies is y₂* and the corresponding explanation models are and where the error terms u₁ and u₂ are jointly bivariate distributed N(0,1). Under the null hypothesis that the error terms are not correlated, ρ equals zero, the model consists of two independent probit equations (Greene 2008, 820). If the correlation coefficient equals zero, the performing arts consumption and movies at the cinema consumption are unrelated (Prieto-Rodriguez and Fernandez-Blanco 2000). The estimation of the equations could be based on classification y₁ = (“daily” or “several times per week” or “several times per month”) = 1 if and y₂ = (“less often” or “never”) = 0, if . In the above example a difference is made between “several times per month” and “less often” but this separation point could be another. With the probit model the marginal effects of each variable could be evaluated (Greene 2008, 821). The marginal effects must be assessed in relation to the zero alternative (like “northern Finland” or “pupil”). The coefficients in the probit model are difficult to interpret since they present what the effect of the variables is on the unobserved dependent variable y*₁. However, the marginal effects of the explanatory variables are on the observed variable y₁. The total marginal effect could be partitioned into two parts: the direct marginal effect and the indirect marginal effect. The latter part is formed through the correlation coefficient of the error terms. The bivariate probability for joint y₁ and y₂ is

(4-14)

where contain all the nonzero elements β₁ and β₂ possibly some zeros in the places of variables in x that appear only in the other equation. The marginal effects of changes in x on this probability are given by

(4-15)

where

where ρ_i* = (2y_i1-1)(2y_i2-1)ρ and w_ij = q_ijz_ij = (2y_ij – 1)x_íjβ_j (Greene 2008, 818-820). If y_i1 = 1 (“daily” or “several times per week” or “several times per month”), 2y_i1-1 = 1 and if y_i1 = 0 (“less often” or “never”) then 2y_i1-1 = -1 for j = 1 (arts) and 2 (movies).