Intermediate Econometrics

• Up to now: responses that are real numbers defined on the whole real line (or when strictly positive real number are coerced to a real number by taking its logarithm)

• Now, responses that are not defined on the whole real line

1. Defined only on a part of the real line
2. Integers or categorical

• The models will be estimated by maximum likelihood: the distribution law of the response will be completely specified and we will estimate the parameters controlling the distributional parameters

\[p_i = F(\eta_i) = F(x^\top_i\beta)\]

• Simplest choice: Identity function for \(F(\cdot)\) \[\implies p_i = \eta_i = x^\top_i\beta\]

• So, the parameter of the Bernoulli distribution is assumed to be a linear function of the covariates

• Advantages:

1. it is very simple to estimate: it is a linear model
2. it can be simply extended to IV estimation

• Being a linear model \(\frac{\partial p_i}{\partial x_{ki}} = \beta_k\), so the estimated parameters can be interpreted as the (constant) marginal effects of the corresponding covariate on the probability of success

• The density functions for the Normal distribution \(\phi(z) = f_Z(z) = \frac{1}{\sqrt{2\pi}} e ^{-\frac{1}{2}z ^ 2}\)
• The density functions for the logistic distribution \(\lambda(z) = f_Z(z) = \frac{e^z}{(1 + e ^ z)^2}\)
• Both density functions are symmetric around 0 and are “bell-shaped”, but they have two important differences

1. the variance of the standard normal distribution is 1 and is equal to \(\pi ^ 2/3\) for the logistic distribution
2. the logistic distribution has much heavier tails than the normal density

• As \(\mu_i=F(\eta_i)\) (with \(\eta = X\beta\)), the marginal effect of the \(k\)-th covariate on the probability is:

\[ \frac{\partial \mu_i}{\partial x_{ik}} = \beta_k f(\eta_i) \]

where \(f\) is the first derivative of \(F\) (the normal and logistic densities, respectively)

• Then, the marginal effect is obtained by multiplying the coefficient by \(f(\eta_i)\) which depends on the value of the covariates for a given observation (the marginal effect is observation-dependent, but the ratio of two marginal effects for two covariates is not, as it is obviously equal to the ratio of the two corresponding coefficients)

• As the coefficient of proportionality is the normal/logistic density, the maximum marginal effect is for \(\eta_i = 0\), which results in a probability of success of 0.5.

• In correspondence of \(\eta_i = 0\) the values of the densities are 0.4 and 0.25 for the normal and logistic densities

• Therefore, a rule of thumb to interpret coefficients is to multiply them respectively by 0.4 and 0.25 for the probit and logit model to get an estimation of the maximum marginal effect

• The coefficients of the logit and probit can therefore not be compared. This is due to the fact that they are scaled differently, as the standard deviation of the logistic distribution is \(\pi/\sqrt{3} \approx 1.81\), compared to 1 for the normal distribution
• Therefore, it would be tempting to multiply the probit coefficients by 1.81 to compare them to the logit coefficients, but, empirically, the value of 1.6 performs better

• As an example, we consider the data set used by Horowitz (1993) which concerns the transport mode chosen for work trips by a sample of 842 individuals in Washington DC in the late sixties

• The response mode is 1 for car and 0 for public transport.

• The covariates are the in- and out-vehicle times (ivtime and ovtime) and the cost differences between car and public transport

• A positive value indicates that car trip is longer/more expensive than the corresponding trip using public transport

• The dataset is mode_choice in the micsr R package
• We make some transformations to the data

1. Dividing the ivtime and ovtime by 60, while the cost by 100
2. We multiply the cost by 8.42 to obtain dollars in 2022 (the CPI for 2022 is 842 with a 100 base in 1967)
3. The generalized cost of a trip is the sum of the monetary cost and the value of the time spent in the transport. We use two-thirds of the minimum hourly wage (about $1.4 in the US in the late sixties, which is about $8 in dollars of 2022) to evaluate an hour of transport

• Models can be fitted via the glm() function

• The coefficient of gcost for the linear-probability model is 0.0226, which means that a one dollar increase of the generalized cost differential will increase the probability of using the car by 2.26 percentage points.

• If we use the previously described rule of thumb to multiply the probit/logit coefficients by 0.4/0.25 in order to have an upper limit for the marginal effect, we get 4.51 and 5.28 percentage points, which are much higher values than for the linear probability model

• This is because the coefficient of the linear model estimates the marginal effect at the sample mean. In our sample, the mean value of the covariate is 2.9

• To get comparable marginal effects for the probit/logit models, we should first compute \(\hat{\beta_1} + \hat{\beta_2} \bar{x}\) (1.15 and 2 respectively for the probit and logit models) and use these values with the relevant densities (\(\phi(1.15) = 0.206\) and \(\lambda(2) = 0.105\))

• At the sample mean, the marginal effects are then \(\phi(1.15)\times 0.1129 = 0.023\) and \(\lambda(2) \times 0.2112 = 0.02255435\): they are therefore very close to the linear probability model coefficient

• We can see the scatterplot and the fitted probability curves

• The fitted probabilities are given by

1. a straight line for the linear probability model
2. an S curve for the probit and logit models. These last two curves are very similar except for low values of the covariate.

• The data set called airbnb has been analyzed by Edelman, Luca, and Svirsky (2017)

• Aim: analyze the presence of racial discrimination on the Airbnb platform

• The authors create guest accounts that differ by the first name chosen. More specifically, the race of the applicant is suggested by the choice of the first name, either a “white” (Emily, Sarah, Greg) or an “African American” (Lakisha or Kareem) first name.

• The response is acceptance and is 1 if the host gave a positive response and 0 otherwise

• We use only three covariates:

1. guest’s race suggested by the first name guest_race
2. the price price (in logs)
3. city, the cities where the experience took place (Baltimore, Dallas, Los Angeles, St. Louis and Washington, DC)

• Note: the mean of the response is \(0.45\) which is a distinctive feature of this data set compared to the previous one (mode_choice)

• As the mean value of the probability of success is close to 50% we can expect that the rule of the thumb which consists of multiplying the logit/probit coefficients by 0.25/0.4 would give an estimated value for the marginal effect close to the one directly obtained in the linear probability model

• The dataset contains some NA values for the covariates we are including in the model: we will simply remove them
  
• We also assign the value 0 to No and 1 to Yes to the outcome variable (acceptance)

• We fit the models (linear probability, logit and probit models)

• We compare the coefficients (those for logit and probit are, respectively, multiplied by 0.25 and 0.4, easing the comparison) and we report the ratio between logit and probit models (showing that is close to 1.6)

• For a 100% increase of the (log-) price, the probability of acceptance reduces by 4.16 percentage points

• The estimated marginal effect for black guests is about -8.5 percentage points (we do not need to compute the derivatives because the covariate is a dummy)

• We should then better compute the difference between the probabilities of acceptance, everything other being equal, which means here for a given price of the property (e.g. The average price being equal to $182) and for the reference city (Baltimore). In the following, we are computing the estimate of

\[P(Y_i=1|x_{i1} = 1, x_{i2} = 1, x_{i3} = \log(mean(price))) - P(Y_i=1|x_{i1} = 1, x_{i2} = 0, x_{i3} = \log(mean(price)))\]

1. For the probit model: \[\Phi(0.536 - 0.213 - 0.116 \times \ln 182) - \Phi(0.536 - 0.116 \times \ln 182) = -0.084\]
2. For the logit model: \[\Lambda(0.859 - 0.341 - 0.187 \times\ln 182) - \Lambda(0.859 - 0.187\times\ln 182) = -0.084\]

• See next slide for the numerical values

• At least in this example, the previous computation of the derivative gives an extremely accurate approximation of the effect of this dummy covariate

• Provide a theoretical foundation to the probit/logit models (we will consider the case where there is a unique covariate)

• We observe that \(y\) is equal to 0 or 1, but we now assume that this values is related to a latent continuous variable (called \(y^*\)) which is unobserved

1. \(y=1\) will result for “high” values of \(y ^ *\)
2. \(y=0\) for low values of \(y ^ *\)

• More specifically we’ll assume that the observation rule is:

\[ \left\{ \begin{array}{rcl} y = 0 & \mbox{if } & y ^ * \leq \psi \\ y = 1 & \mbox{if } & y ^ * > \psi \\ \end{array} \right. \]

where \(\psi\) is an unknown threshold

• Let’s assume that the value of \(y^ *\) is partly explained by an observable covariate \(x\)
• The unexplained part being modelized by a random error \(\epsilon\)
• We then have: \(y ^ * = \beta_1 + \beta_2 x + \epsilon\), so that the observation rule becomes:

\[ \left\{ \begin{array}{rcl} y = 0 & \mbox{if } & \epsilon \leq \psi - \beta_1 - \beta_2 x\\ y = 1 & \mbox{if } & \epsilon > \psi - \beta_1 - \beta_2 x \\ \end{array} \right. \]

• This observation rule depends on \(\psi - \beta_1\) and not on the separate values of \(\psi\) and \(\beta_1\)
• Therefore, \(\psi\) can be set to any arbitrary value, for example 0

• Then, the probability of success is: \(1 - F(-\beta_1 - \beta_2 x)\) where \(F\) is the cumulative density function of \(\epsilon\)
• For example, if \(\epsilon \sim \mathcal{N} (0, \sigma^2)\), then \(\mbox{P}(Y = 1 \mid x) = 1 - \Phi(-(\beta_1 + \beta_2 x) / \sigma)\)

1. We can see from this expression that only \(\beta_1 / \sigma\) and \(\beta_2 / \sigma\) can be identified, so that, \(\sigma\) can be set to any arbitrary value, for example 1
2. Moreover, by the symmetry of the normal distribution, we have \(1 - \Phi(-z) = \Phi(z)\), so that the probability of success becomes \[\mbox{P}(Y = 1 \mid x) = \Phi(\beta_1 + \beta_2 x)\] which defines the probit model

• Assuming that the distribution of \(\epsilon\) is logistic, we have a probability of success equal to \(\mbox{P}(Y = 1 \mid x) = 1 - \Lambda(- \beta_1-\beta_2 x)\) which reduces, as the logistic distribution is also symmetric, to: \[\mbox{P}(Y = 1 \mid x) = \Lambda(\beta_1 + \beta_2 x) = e^{\beta_1 + \beta_2 x} / (1 + e ^ {\beta_1 + \beta_2 x})\]

• Consider now that we can define a utility function for the two alternatives that correspond to the two values of the binary response
• As an example, \(y\) equals 1 or 0 if car or public transport is chosen, and the only covariate \(x\) is the generalized cost
• The utility of choosing a transport mode doesn’t depend only on the generalized cost, but also on some other unobserved variables. The effect of these variables are modelized as the realization of a random variable \(\epsilon\)
• We can therefore define the following random utility functions:

\[ \left\{ \begin{array}{rcl} U_0 &=& \beta_{01}+ \beta_2 x_0 + \epsilon_0 \\ U_1 &=& \beta_{11} + \beta_2 x_1 + \epsilon_1 \end{array} \right. \]

where \(\beta_2\) is the marginal utility of $1.

• The choice of the individual is deterministic: They will choose the car if the utility of this mode is greater than the utility of public transport
• Then, we have the following observation rule:

\[ \left\{ \begin{array}{rcl} y = 0 & \mbox{if } & \epsilon_1 - \epsilon_0 \leq - (\beta_{11} - \beta_{01}) - \beta_2 (x_1 - x_0)\\ y = 1 & \mbox{if } & \epsilon_1 - \epsilon_0 > - (\beta_{11} - \beta_{01}) - \beta_2 (x_1 - x_0)\\ \end{array} \right. \]

• Denoting \(\epsilon = \epsilon_1 - \epsilon_0\) the error difference, \(\beta_1 = \beta_{11} - \beta_{01}\) and \(x = x_1 - x_0\) the difference of generalized cost for the two modes, we have:

\[ \left\{ \begin{array}{rcl} y = 0 & \mbox{if } & \epsilon \leq - (\beta_1 + \beta_2 x)\\ y = 1 & \mbox{if } & \epsilon > - (\beta_1 + \beta_2 x)\\ \end{array} \right. \]

• The probability of “success” (here choosing the car) is therefore \(\mbox{P}(Y = 1 \mid x) = 1 - F(- \beta_1 - \beta_2 x)\), with \(F\) as the cumulative density of \(\epsilon\). If the distribution is symmetric, this probability reduces once again to \(\mbox{P}(Y = 1 \mid x) = F(\beta_1 + \beta_2 x)\), and the probit or logit models are obtained by choosing the normal or the logistic distribution, respectively

• The exponential family is defined by the following density function:

\[ f(y;\theta,\phi) = e ^ {\displaystyle\left(y\theta - b(\theta)\right)/\phi + c(y, \phi)} \]

with \(\theta\) and \(\phi\) being respectively a position and a scale parameter

• Linear models are a specific case of generalized linear models with a normal distribution and an identity link

• We have in this case the following density function:

\[ \phi(y;\mu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma} e ^ {-\frac{1}{2}\frac{(y - \mu)^2}{\sigma ^ 2}}= e^{\frac{y\mu - 0.5 \mu ^ 2}{\sigma ^ 2}- 0.5 y ^ 2 / \sigma ^ 2 - 0.5 \ln(2\pi\sigma ^ 2)} \]

which is a member of the exponential family with \(\theta = \mu\), \(\phi = \sigma^ 2\), \(b(\theta) = \theta ^ 2/2\) and \(c(y, \phi) = -(y ^ 2 / \phi + \ln(2\pi\phi))/2\)

• For the Gaussian model with an identity link, we have, for a given set of estimates (which leads to the proposed model): \(\hat{\mu}_i = \hat{\eta}_i = x^\top_i \hat \beta\) and the log-likelihood function is:

\[ \ln L(y, \hat{\mu}) = - \frac{n}{2}\ln(2 \pi + \sigma ^ 2) - \frac{1}{2\sigma ^2} \sum_{i=1} ^ n (y_i - \hat{\mu}_i) ^ 2 \]

• For a hypothetical “perfect” or saturated model with a perfect fit, we would have \(\hat{\mu}_i = y_i\), so that the log-likelihood would be \(- \frac{n}{2}\ln(2 \pi + \sigma ^ 2)\)

• Minus two times the difference of these two values of the log likelihood function is called the scaled deviance of the proposed model: \[ D^*(y;\hat\mu) = \sum_{i=1} ^ n\frac{(y_i - \hat{\mu}_i) ^ 2}{\sigma ^ 2} \]

• The deviance is obtained by multiplying the scaled deviance by \(\sigma ^ 2\)

\[ D(y; \hat{\mu}) = \sum_{i=1} ^ n(y_i - \hat{\mu}_i) ^ 2 \]

which is simply, for the linear model, the sum of square residuals

• The probability mass function is given by the probability of success \(\mu\) (also denoted with \(p\) previously) if \(y = 1\) and by the probability of failure \(1 - \mu\) if \(y=0\)
• This probability can be compactly written as \[f_Y(y; \mu) = P(Y=y; \mu) = \mu ^ y (1 - \mu) ^ {1 - y}\] or as:

\[ f(y;\mu) = e ^ {y \ln \mu + (1 - y) \ln(1 - \mu)}=e ^ {y \ln \frac{\mu}{1 - \mu} + \ln(1 - \mu)}= e^{y\theta - \ln (1 + e ^ \theta)} \]

which is a member of the exponential family with:

1. \(\theta = \ln\frac{\mu}{1 -\mu}\)
2. \(b(\theta) = \ln(1 + e ^ \theta)\) \(c(\theta,y) = 0\)
3. \(\phi=1\)

• The model is fully characterized once the link is specified

• For the logit model, we have \(\mu = \frac{e ^ \eta}{1+e ^ \eta}\), so that \(\eta = \ln \frac{\mu}{1 - \mu} = g(\mu)\)

• We then have \(\theta = \eta\), so that the logit link is called the canonical link for binomial models

• As the density for the binomial model returns a probability, the log-likelihood for the saturated model is zero. Therefore, the deviance is:

\[ D(y;\hat{\mu}) = 2 \sum_{i=1} ^ n \left(y_i \ln \hat{\mu}_i + (1 - y_i) \ln(1 - \hat{\mu}_i)\right) \]

• The null model is a model with only an intercept: in this case, \(\hat{\mu}_i = \hat{\mu}_0\) and the maximum likelihood estimate of \(\mu_0\) is \(\sum_{i=1} ^ n y_i / n\), i.e., the share of success in the sample

• The deviance of this model is called the null deviance

• An alternative to the deviance as a measure of the fit of the model is the generalized Pearson statistic, defined as:

\[ X ^ 2 = \sum_{i=1} ^ n \frac{(y_i - \hat{\mu}_i) ^ 2}{{\hat V} (\hat{\mu}_i)}= \sum_{i=1} ^ n \frac{(y_i - \hat{\mu}_i) ^ 2}{\hat{\mu}_i(1 - \hat{\mu}_i)} \]

• In the linear model, residuals have several interesting properties:

1. they are homoscedastic (or at least they may be homoscedastic if the variance of the conditional distribution of the response is constant)
2. they have an intuitive meaning, as they are the difference between the actual and the fitted values of the response, the latter being an estimate of the conditional mean of the response
3. they are related to the value of the objective function, which is the sum of square residuals

• The most obvious definition of the residuals for binomial models is the response residuals, which are simply the difference between the response and the prediction of the model (the fitted probability of success \(\hat{\mu}\))

• However, these residuals (\(y_i - \hat{\mu}_i\)) are necessarily heteroscedastic, as the variance of \(Y_i\) is \(\mu_i(1 - \mu_i)\)

• Scaling the response residuals by their standard deviation leads to Pearson’s residuals: \[\frac{y_i - \hat{\mu}_i}{\sqrt{\hat{\mu}_i(1 - \hat{\mu}_i)}}\]

• The sum of squares of Pearson’s residuals is the generalized Pearson statistic

• The deviance residuals are such that the sum of their squares equals the deviance statistic \(D\)
• They are defined by: \[(2 y_i - 1) \sqrt{2}\sqrt{y_i \ln \hat{\mu}_i + (1 - y_i) \ln (1 - \hat{\mu}_i)}\]

the term \(2 y_n - 1\) gives a positive sign for the residuals of observations for which \(y_n = 1\) and a negative sign for \(y_n = 0\), as for the two other types of residuals

• We have already seen that the R routine to fit a logit/probit model is based on the glm() function. When fitting a generalized linear model, three models are considered:

1. the saturated model, for which there is one parameter for every observation and a perfect fit; therefore the log-likelihood, the deviance and the number of degrees of freedom are 0
2. the null model, with only one estimated coefficient and \(n-1\) degrees of freedom
3. the proposed model, with \(p\) estimated parameters, and therefore \(n - p\) degrees of freedom.

• Let’s see the summary output: The output indicates the deviance of the null and the proposed model, along with their respective degrees of freedom (The latter is called the residual deviance)

• The residual deviance can be extracted accessing to the glm object

• The null deviance can be extracted accessing to the glm object, as well
• We can see that is equivalent to fit a model only including an intercept

• The residuals can be extracted from the fitted model using resid
• The resid method for glm objects has a type argument which can be equal to the following three options

1. “response”
2. “pearson”
3. “deviance”

• However, the most interesting plot to consider is based on the binned residuals (not shown here)

• The fitted values of the model can be expressed on the scale of the linear predictor or the response

• The stats::glm function uses an iterative weighted least squares method to fit GLMs.
• Probit and logit models are usually estimated by maximum likelihood
  
• Given the log-likelihood, the log-likelihood derivatives (up to the second order are obtained) and they are used in algorithms (like Fisher-Scoring or Newton-Raphson) to obtain the maximum likelihood estimates of the model parameters.
• The second order derivative is used for computing the (estimated) standard errors of the regression coefficient estimators, using the asymptotic distribution of the maximum likelihood estimator
• We do not go in the details, but it is important to know that other estimators of the variance-covariance matrix of the regression coefficients can be used

• Once several models are estimated, the evaluation and the selection process of one of them can be based on several indicators

1. The value of the objective function, which is the log-likelihood
2. Closely related to the log-likelihood is the deviance, which is the opposite of twice the log-likelihood

• These measures usually indicate an important difference between the constrained and unconstrained model

• However, the comparison between the constrained and unconstrained models is spurious, because adding further covariates, even if they are irrelevant, necessarily increases the fit of the model

• Therefore, it is important to consider indicators that penalize highly parametrized models. The two most popular indicators are the Akaike and the Bayesian information criteria (AIC and BIC):

1. \(\mbox{AIC} = - 2 \ln L + 2 p\)
2. \(\mbox{BIC} = - 2 \ln L + p \ln n\).

• They are obtained by augmenting the deviance by a term which is a multiple of the number of fitted parameters

• The rule being to select the model for which the statistic is lower (lower is the best)

• Let’s consider the example of the mode_choice:

1. we assign to \(y\) and to \(X\), respectively, the outcome and the model matrix
2. we implement the log-likelihood
3. we set the starting values
4. we optimize the log-likelihood using numerical optimization (minimizing the negative log-likelihood)
5. we analyze the estimate and compare with those obtained via the glm function
6. we extract the standard errors using the observed information matrix (the negative of the hessian matrix) and compare them with those obtained via the glm function
7. we analyze the log-likelihood in the maximum
8. we obtaine the deviance and the AIC

1. we assign to \(y\) and to \(X\), respectively, the outcome and the model matrix
2. we implement the log-likelihood

3. we set the starting values
4. we optimize the log-likelihood using numerical optimization (minimizing the negative log-likelihood)
5. we analyze the estimate and compare with those obtained via the glm function ::

6. we extract the standard errors using the observed information matrix (the negative of the hessian matrix) and compare them with those obtained via the glm function

7. we analyze the log-likelihood in the maximum
8. we obtain the deviance and the AIC

• Let’s consider the logit and probit regression model to the mode choice data and we compare the specifications including
  

1. the generalized cost (gcost)
2. the monetary cost (cost), in- and out-vehicle time (ivtime and ovtime)

• Let’s consider the logit and probit regression model to the mode choice data and we compare the specifications including
  

1. the generalized cost (gcost)
2. the monetary cost (cost), in- and out-vehicle time (ivtime and ovtime)

• To test hypotheses involving nested models, three asymptotically equivalent tests are commonly used:

1. Likelihood Ratio (LR) Test: The LR test is based on the comparison of the maximized log-likelihoods of the restricted (R) and unrestricted (U) models:

\[LR=2(\ell(\hat \beta^U)−\ell(\hat \beta^R))\]

2. Wald Test: The Wald test relies only on the unrestricted estimator and tests whether the parameter restrictions are satisfied:

\[W= (\hat \beta^U − \beta_0)^\top Var(\hat \beta^U)^{−1} (\hat \beta^U−\beta_0)\]

where \(\beta_0\) satisfies the null restrictions (e.g., \(\beta_0=0\))

• Score Test: The Score test is based only on the restricted model and evaluates whether the slope of the log-likelihood at the restricted estimate is significantly different from zero:

\[Score =S(\hat \beta^R)^\top I(\hat \beta^R)^{−1} S(\hat \beta^R)\]

where \(S(β)=\partial \ell (\beta)/\partial \beta\) is the score vector and \(I(\beta)\) is the Fisher information matrix (the expected value of the negative hessian)

• Under the null hypothesis, all three test statistics converge in distribution to a \(\chi^2(q)\) where q is the number of restrictions

• We estimated a model with the generalized cost (\(g\)) as a unique covariate, which was computed as: \(g_i = c_i + 8(i_i + o_i)\), where \(c\), \(i\), and \(o\) are the differences in monetary cost, in-vehicle time and out-vehicle time, based on the hypothesis that time value was $8 per hour

• The unconstrained model for the probit case is: \[ P(Y_i = 1) = \Phi(\beta_0 + \beta_c c_i + \beta_i i_i + \beta_o o_i) \]

• The constrained model implies the two following hypotheses: \(H_0: \beta_o = \beta_i = 8 \beta_c\). It is more convenient to rewrite the model so that, under \(H_0\), a subset of the parameters are 0:

\[ \begin{array}{rcl} P(Y_i = 1) &=& \Phi\left(\beta_0 + \beta_c \left(c_i + 8 (i_i + o_i)\right) + (\beta_i - 8\beta_c)i_i + (\beta_o - 8 \beta_c) o_i\right)\\ &=& \Phi(\beta_0 + \beta_c g_i + \beta_i'i_i + \beta_o'o_i) \end{array} \]

where \(\beta_i' = (\beta_i - 8\beta_c)\) and \(\beta_o' = (\beta_o - 8\beta_c)\) are the reduced form parameters of the binomial regression with the generalized cost, the in-vehicle and out-vehicle time as covariates

• With this parametrization, the set of hypotheses is simply \(\beta_i' = \beta_o' = 0\)

• The two statistics are very close and in both cases we are rejectung the null hypothesis of

• We can use three test (included in the micsr package) to assess

1. the normality of latent error term (under probit)
2. the homoscedasticity assumption (of the latent error term)
3. if we are missing non-linear terms or omitted variables

• Our probit model seems to be correctly specified, as the three hypotheses are not rejected