Econometrics Midterm PDF

Econometrics 2 1. Qualitative and Limited Dependent Variable Models Qualitative and Limited Dependent Variable Models Qualitative: dependent variable is in a form of alternatives. Limited: dependent variables are in continuous form; however, their values are not completely observable. Economic Appli...

Econometrics 2 1. Qualitative and Limited Dependent Variable Models Qualitative and Limited Dependent Variable Models Qualitative: dependent variable is in a form of alternatives. Limited: dependent variables are in continuous form; however, their values are not completely observable. Economic Application Example: An economic model explaining why some individuals take a second or third job and engage in “moonlighting”. 1 if it engages with moonlighting 𝑦𝑖 = { 0 otherwise An economic model of why some legislators in the U.S. House of Representatives vote for a particular bill and others do not. An economic model explaining why some loan applications are accepted and others are not at a large metropolitan bank. An economic model explaining why some individuals vote for increased spending in a school board election and others vote against. An economic model explaining why some female college students decide to study engineering and others do not. Binary Choice Models Review on Regression Model – Interest on E(𝑦|𝑥) Let 𝑥𝑖 = (𝑥𝑖0 = 1, 𝑥𝑖1 , 𝑥𝑖2 , … , 𝑥𝑖𝑘 ) with 𝑘 number of independent variables Then P(𝑦𝑖 = 1|𝑥𝑖 ) = p(𝑥𝑖 ) and P(𝑦𝑖 = 0|𝑥𝑖 ) = 1 − p(𝑥𝑖 ) (complementary probability) Proof: The conditional probability function for 𝑦𝑖 is 𝑓(𝑦𝑖 |𝑥𝑖 ) = p(𝑥𝑖 )𝑦𝑖 (1 − p(𝑥𝑖 ))1−𝑦𝑖 where 𝑦𝑖 = {0,1} Review on Regression Model – Interest on E(𝑦|𝑥) Let 𝑥𝑖 = (𝑥𝑖0 = 1, 𝑥𝑖1 , 𝑥𝑖2 , … , 𝑥𝑖𝑘 ) with 𝑘 number of independent variables Then P(𝑦𝑖 = 1|𝑥𝑖 ) = p(𝑥𝑖 ) and P(𝑦𝑖 = 0|𝑥𝑖 ) = 1 − p(𝑥𝑖 ) (complementary probability) Proof: The conditional probability function for 𝑦𝑖 is 𝑓(𝑦𝑖 |𝑥𝑖 ) = p(𝑥𝑖 )𝑦𝑖 (1 − p(𝑥𝑖 ))1−𝑦𝑖 where 𝑦𝑖 = {0,1} 𝑓(𝑦𝑖 |𝑥𝑖 ) = p(𝑥𝑖 )𝑦𝑖 (1 − p(𝑥𝑖 ))1−𝑦𝑖 where 𝑦𝑖 = 0,1 Following this, 𝑓(1|𝑥𝑖 ) = p(𝑥𝑖 )1 (1 − p(𝑥𝑖 ))1−1 𝑓(0|𝑥𝑖 ) = p(𝑥𝑖 )0 (1 − p(𝑥𝑖 ))1−0 𝑓(1|𝑥𝑖 ) = p(𝑥𝑖 )1 (1 − p(𝑥𝑖 ))0 𝑓(0|𝑥𝑖 ) = (1 − p(𝑥𝑖 ))1 𝑓(1|𝑥𝑖 ) = p(𝑥𝑖 ) 𝑓(0|𝑥𝑖 ) = 1 − p(𝑥𝑖 ) Now, how are we going to approximate P(𝑦𝑖 = 1|𝑥𝑖 ) (or 𝑓(1|𝑥𝑖 ) or p(𝑥𝑖 ))? Transportation Economics Case: How do individuals decide between driving and commuting to work? Assumption: there are only two alternatives Possible factors Individual characteristics: age, sex, income Automobile characteristics: reliability, comfort, fuel economy Public transportation characteristics: reliability, cost, safety Single factor: commuting time as 𝒙𝒊 𝑥𝑖 = commuting time by bus – commuting time by car, for the 𝑖 𝑡ℎ individual 𝑥𝑖 = 𝑡𝑖𝑚𝑒𝑏𝑢𝑠,𝑖 − 𝑡𝑖𝑚𝑒𝑐𝑎𝑟,𝑖 a priori: we expect that as 𝑥𝑖 increases, commuting time by bus is greater than commuting time by car, holding all else constant, an individual would be more inclined to drive. Positive relationship between the difference of commuting time and probability that an individual will drive to work: increase in 𝑥𝑖 → increase in p(𝑦𝑖 = 1|𝑥𝑖 ) Qualitative and Limited Dependent Variable Models A. Dummy Dependent Variable Linear Probability Model (LPM) Conditional Expected value E(𝑦𝑖 |𝑥𝑖 ) = ∑1𝑖=0 𝑥𝑖 p(𝑥𝑖 ) E(𝑦𝑖 |𝑥𝑖 ) = 0 ∙ 𝑓(0|𝑥𝑖 ) + 1 ∙ 𝑓(1|𝑥𝑖 ) E(𝑦𝑖 |𝑥𝑖 ) = 0 ∙ (1 − p(𝑥𝑖 )) + 1 ∙ p(𝑥𝑖 ) E(𝑦𝑖 |𝑥𝑖 ) = p(𝑥𝑖 ) Following this, p(𝑥𝑖 ) = E(𝑦𝑖 |𝑥𝑖 ) = 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 = 𝛽0 + 𝛽1 𝑥𝑖1 + 𝛽2 𝑥𝑖2 + ⋯ + 𝛽𝑘 𝑥𝑖𝑘 Let 𝑒𝑖 the random error (observed outcome – conditional mean) 𝑒𝑖 = 𝑦𝑖 − E(𝑦𝑖 |𝑥𝑖 ) Then, 𝑦𝑖 = E(𝑦𝑖 |𝑥𝑖 ) + 𝑒𝑖 𝑦𝑖 = 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 + 𝑒𝑖 and E(𝑒𝑖 |𝑥𝑖 ) = 0 (least square estimator parameters are unbiased) E(𝑒𝑖 𝑥𝑖 ) = 0 (least square estimator parameters are consistent) Marginal effect of LPM For a continuous variable 𝑥𝑖𝑗 , 𝑗 = 1,2, … , 𝑘 the marginal effect of LPM is given by 𝜕E(𝑦𝑖 |𝑥𝑖 ) 𝑀𝐸𝐿𝑃𝑀 = 𝛽𝑗 = 𝜕𝑥𝑖𝑗 Problems of LPM: 1. Logical inconsistencies. Suppose 𝛽𝑗 > 0, then we represent A unit increase in 𝑥𝑖𝑗 leads to 𝛽𝑗 increase in p(𝑥𝑖 ) where 𝛽𝑗 is constant (slope of a line), which means there is a possibility of greater than 1 probability when 𝑥𝑖𝑗 gets larger. However, recall that we can only have 0 ≤ p(𝑥𝑖 ) ≤ 1. 2. The 𝑦𝑖 , E(𝑦𝑖 |𝑥𝑖 ) and 𝑒𝑖 take only two values: 𝑦𝑖 = {0, 1} E(𝑦𝑖 |𝑥𝑖 ) = {0, 1} 𝑒𝑖 = {−E(𝑦𝑖 |𝑥𝑖 ), 1 − E(𝑦𝑖 |𝑥𝑖 )} Why? For 𝑦𝑖 = {0,1}, then 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 = {0,1} That is, if 𝑦𝑖 = 1, then 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 = 1 and 𝑦𝑖 = 0, then 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 = 0 2. The 𝑦𝑖 , E(𝑦𝑖 |𝑥𝑖 ) and 𝑒𝑖 take only two values: 𝑦𝑖 = {0, 1} E(𝑦𝑖 |𝑥𝑖 ) = {0, 1} 𝑒𝑖 = {−E(𝑦𝑖 |𝑥𝑖 ), 1 − E(𝑦𝑖 |𝑥𝑖 )} Implication on the random error 𝑒𝑖 For 𝑦𝑖 = 0, then 𝑒𝑖 = 0 − (𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 ) 𝑒𝑖 = −𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 and 𝑦𝑖 = 1, then 𝑒𝑖 = 1 − 𝛽0 + ∑𝑘𝑗=1 𝛽𝑗 𝑥𝑖𝑗 3. Conditional variance in the random error is heteroskedastic. 𝑣ar(𝑒𝑖 |𝑥𝑖 ) = p(𝑥𝑖 )[1 − p(𝑥𝑖 )] = 𝜎𝑖2 To solve the problems of LPM, it needs transformation so that 𝑝(𝑥𝑖 ) will lie on the interval [0,1]. One of the suitable distributions for this is cumulative normal function. Probit Model The probability distribution p(𝑥𝑖 ) = 𝑃𝑖 = F(𝑍𝑖 ) then the probit model is given by 𝑘 𝑍𝑖 = 𝛽0 + ∑ 𝛽𝑗 𝑥𝑖𝑗 + 𝑒𝑖 𝑗=1 𝑍 is the standard normal distribution with mean of 0 and variance of 1, F(∙) is the cumulative density function using the standardized 𝑍 table. Marginal effect of Probit Model The marginal effect of variable 𝑥𝑖𝑗 is given by 𝜕𝑃𝑖 = 𝛽𝑗 ∙ F(𝑍𝑖 ) 𝜕𝑥𝑖𝑗 𝜕𝑃𝑖 where is the marginal effect of variable 𝑥𝑖𝑗 , 𝜕𝑥𝑖𝑗 𝛽𝑗 is the parameter of variable 𝑥𝑖𝑗 F(𝑍𝑖 ) is the cumulative probability computed at the means of 𝑥𝑖𝑗 Comparison: Probit and Logit Models The cumulative logistic distribution is used as the underlying distribution for the model. Logit Model Logistic distribution: 𝑃𝑖 = 1 𝑃 → 𝑍𝑖 = ln 1−𝑃𝑖 1+𝑒 −𝑧𝑖 𝑖 then, the logit model is given by 𝑘 𝑃𝑖 ln = 𝛽0 + ∑ 𝛽𝑗 𝑥𝑖𝑗 + 𝑒𝑖 1 − 𝑃𝑖 𝑗=1 where 𝑃𝑖 is the probability of an event happening, 1 − 𝑃𝑖 is the probability of an event not happening. Marginal effect of Logit Model The marginal effect of variable 𝑥𝑖𝑗 is given by 𝜕𝑃𝑖 = 𝛽𝑗 𝑃𝑖 (1 − 𝑃𝑖 ) 𝜕𝑥𝑖𝑗 𝜕𝑃𝑖 where is the marginal effect of variable 𝑥𝑖𝑗 , 𝜕𝑥𝑖𝑗 1 𝑃𝑖 = 1+𝑒 −𝑧𝑖 is the probability, 𝑍𝑖 is computed at the means of 𝑥𝑖𝑗 Estimation methods Ordinary Least Squares (OLS) – linear The ordinary least square minimizes the square of the residuals. Maximum Likelihood Estimation (MLE) – probabilistic The maximum likelihood estimation method maximizes the probability of observing the dataset given a model and its parameters. Model comparison Pseudo R2: goodness of fit of the model Pseudo R2 fit statistics for generalized linear models take on similar values to their ordinary least squares counterparts but are based on maximum likelihood estimates instead of sums of squares. In general, higher values indicate that the model is better at discriminating. Information Criterion: estimation of prediction error Information Criterion is a method used to select the best model from a set of models by maximizing the likelihood of the data while penalizing the number of parameters to prevent overfitting. Akaike Information Criterion (AIC) or Bayes Information Criterion (BIC) AIC is an estimate of a constant plus the relative distance between the unknown true likelihood function of the data and the fitted likelihood function of the model, whereas BIC is an estimate of a function of the posterior probability of a model being true, under a certain Bayesian setup. Multinomial Choices In probit and logit models, the decision maker chooses between two alternatives. But what if the choices involve more than two alternatives? Multinomial choice economic application examples: If you are shopping for a laundry detergent, which one do you choose? Tide, Ariel, Breeze, Surf, and so on. The consumer is faced with a wide array of alternatives. Marketing researchers relate these choices to prices of the alternatives, advertising, and product characteristics. If you enroll in business school, will you major in economics, marketing, management, finance, or accounting? 0 − economics 1 − marketing 𝑦𝑖 = 2 − management 3 − finance { 4 − accounting If you are going to a mall on a shopping spree, which mall will you go to, and why? When you graduated from high school, you had to choose between not going to college and going to a private four-year college, a public four-year college, or a two-year college. What factors led to your decision among these alternatives? Multinomial Logit Model Multinomial logit is a technique used when there are more than 2 categories for the dummy variable: 𝑘 E(𝑌𝑖 ∗ ) = 𝛽0 + ∑ 𝛽𝑗 𝑋𝑖𝑗 = 𝑍𝑖 𝑗=1 where 𝑌 ∗ is an index with values = 0, 1, 2, 3, … Unordered Categories Adopt different technologies without ranking the preference Ordered Categories (Ordered Logit Model) High, Medium, Low Dummy variables with 𝑚 categories require the calculations of 𝑚 − 1 equations Reference category – usually the one with most frequency probability of the event happening 𝑃(𝑌𝑖 =𝑚) ln = 𝛽𝑚 + ∑𝑘𝑗=1 𝛽𝑚𝑗 𝑋𝑖𝑗 = 𝑍𝑚𝑖 𝑃(𝑌𝑖 =0) probability of the event not happening Computation of Probabilities for Unordered Logit Reference category – usually the one with most frequency 1 𝑃(𝑌𝑖 = 0) = 1+ ∑𝑚 ℎ=1 𝑒 𝑍ℎ𝑖 For 𝑚 = 1, 2,3,4, … 𝑒 𝑍𝑚𝑖 𝑃(𝑌𝑖 = 𝑚) = 1 + ∑𝑚ℎ=1 𝑒 𝑍ℎ𝑖 Computation of Probabilities for Ordered Logit Generally, for 𝑚 ordered categories: 𝑒 𝑍𝑖 −𝜅𝑤 𝑃(𝑌𝑖 > 𝑤) = , 𝑤 = 1,2, … , 𝑚 − 1 1 + 𝑒 𝑍𝑖 −𝜅𝑤 which implies 𝑒 𝑍𝑖 −𝜅1 𝑃(𝑌𝑖 = 0) = 1 − 1 + 𝑒 𝑍𝑖 −𝜅1 𝑒 𝑍𝑖 −𝜅𝑤−1 𝑒 𝑍𝑖 −𝜅𝑤 𝑃(𝑌𝑖 = 𝑤) = − , 𝑤 = 1, 2,3, … , 𝑚 − 1 1 + 𝑒 𝑍𝑖 −𝜅𝑤−1 1 + 𝑒 𝑍𝑖 −𝜅𝑤 𝑒 𝑍𝑖 −𝜅𝑚−1 𝑃(𝑌𝑖 = 𝑚) = 1 + 𝑒 𝑍𝑖 −𝜅𝑚−1 Computation of Probabilities for Ordered Logit Consider 3 ordered categories: 1 𝑃(𝑌𝑖 = 0) = 1 + 𝑒 𝑍𝑖 −𝜅1 1 1 𝑃(𝑌𝑖 = 1) = − 1 + 𝑒 𝑖 2 1 + 𝑒 𝑍𝑖 −𝜅1 𝑍 −𝜅 𝑚−1 1 𝑃(𝑌𝑖 = 2) = 1 − ∑ 1 + 𝑒 𝑍𝑖 −𝜅𝑤 𝑤=1 where 𝑍 is the predicted value for the ordered logit regression 𝜅1 is the first cut-off or threshold value 𝜅2 is the second cut-off or threshold value Comparison of Probability Estimates using Multi-level Logit Regressions Note: Observations on the extreme sides of the distribution will differ depending on whether it is ordered or unordered. Limited Dependent Variable: Tobit Model In probit, logit, and multinomial logit, the dependent variable 𝑌 is a dummy variable. The dependent variable 𝑌 for Tobit is a censored or latent variable which is not observed. In the standard Tobit model (Tobin 1958), the dependent variable y is left- censored at zero: 𝑘 𝑌𝑖 ∗ = 𝛽0 + ∑ 𝛽𝑗 𝑋𝑖𝑗 + 𝜀𝑖 if 𝑌𝑖 ∗ > 0 𝑌𝑖 = { 𝑗=1 0 if 𝑌𝑖 ∗ ≤ 0 where 𝑌𝑖 ∗ is observed if 𝑌𝑖 ∗ > 0 and unobserved if 𝑌𝑖 ∗ ≤ 0 Generally, the dependent variable can be either left-censored, right-censored, or both left-censored and right-censored, where the lower and/or upper limit of the dependent variable can be any number: 𝑎 if 𝑌𝑖 ∗ ≤ 𝑎 𝑘 𝑌𝑖 = 𝑌𝑖 ∗ = 𝛽0 + ∑ 𝛽𝑗 𝑋𝑖𝑗 + 𝜀𝑖 if 𝑎 < 𝑌𝑖 ∗ < 𝑏 𝑗=1 {𝑏 if 𝑌𝑖 ∗ ≥ 𝑏 where 𝑎 is the lower limit and 𝑏 is the lower limit of the dependent variable 𝑌𝑖. Computation of Probability of 𝑌𝑖 = 0 𝑍𝑖 𝑃𝑖 = 𝐹 (− ) 𝜎 which is similar to probit but the 𝑍-value is divided by the standard deviation 𝜎. Marginal Effect of Tobit Model The marginal effect of variable 𝑥𝑖𝑗 is given by 𝜕𝐸(𝑦𝑖 |𝑥𝑖 ) 𝛽1 + 𝛽2 𝑥𝑖 = 𝛽2 ∙ F ( ) 𝜕𝑥𝑖 𝜎 where F is the cumulative distribution function (cdf) of the standard normal random variable that is evaluated at the estimates and a particular x-value. Because the cdf values are positive, the sign of the coefficient tells the direction of the marginal effect, but the magnitude of the marginal effect depends on both the coefficient and the cdf. If 𝛽2 > 0, as 𝑥 increases, the cdf function approaches one, and the slope of the regression function approaches that of the latent variable model (see figure in the next slide). The marginal effect can be decomposed into two factors called the ‘‘McDonald-Moffit’’ decomposition: 𝜕𝐸(𝑦𝑖 |𝑥𝑖 ) 𝜕𝐸(𝑦𝑖 |𝑥𝑖 , 𝑦 > 0) 𝜕𝑃(𝑦𝑖 > 0) = 𝑃(𝑦𝑖 > 0) ∙ + 𝐸(𝑦𝑖 |𝑥𝑖 , 𝑦 > 0) ∙ 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑥𝑖 The first factor accounts for the marginal effect of a change in 𝑥 for the portion of the population whose 𝑦 -data is observed already. The second factor accounts for changes in the proportion of the population who switch from the 𝑦 -unobserved category to the y- observed category when 𝑥 changes. Lecture 2. Dynamic Models Time-Series Data Time-series data are data collected over time on one particular economic unit. * Economic unit may refer to individuals, households, firms, countries, etc. Two (2) features of time-series data: Modeling Dynamic Relationships 1. These time-series observations are Introducing lagged variables into the more likely to be correlated by their past model. values. The model depends on who takes the 2. It has natural ordering according to form of lagged values. time. Shuffling the observations may present a danger of confounding the possible existence of dynamic-evolving relationships between variables. Distributed Lag Model DL(q) Suppose the value of a variable 𝑦 depends on current and past values of another variable 𝑥, up to 𝑞 periods into the past. That is, where: The equation above is called distributed lag model; 𝛽𝑎 are called the distributed lag weights at lag 𝑎; and  𝛽0, 𝛽1, 𝛽2, … is called a 3. No feedback effects: Cov(𝑌𝑡−𝑘 , 𝑋𝑡 )= 0 distributed lag pattern. for all 𝑘 > 0 If the effect of 𝑥 on 𝑦 cuts off after 𝑞, Changes in 𝑌 do not influence future then it is a finite distributed lag. values of 𝑋 Distributed Lag Model 𝐃𝐋 (𝐪 ) − 4. Normality of error 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐢𝐞𝐫𝐬 terms: ∈𝑡 ~𝑁(0, 𝜎 2 ) Zero mean of errors: E(∈𝑡 ) = 0 Homoscedasticity: Var(∈𝑡 ) = 𝜎 2 5. No autocorrelation in the residuals: Cov 𝜖𝑡 , 𝜖𝑠 ) = 0 𝑡 ≠ 𝑠 Impact multiplier 𝜷𝟎 Stationarity Assumption  represents the immediate effect of 𝑋 on 𝑌. The variables used in the models are stationary – mean, variance, and 𝒂 –period delay multiplier 𝜷𝒂 autocovariance are constant over time.  represents how a past change in 𝑋 affects the 𝑌. Interim multiplier ∑𝒂𝒋=𝟎 βj where 𝑎 < 𝑞  the cumulative effect of 𝑋 on 𝑌 over multiple periods. Total multiplier ∑𝒂𝒋=𝟎 βj where 𝑎 = 𝑞 Possible Use of Distributed Lag Model 𝐃𝐋(𝐪)  represents the long-term cumulative effect of 𝑋 on 𝑌. Can be used for forecasting. Assumptions of the Distributed Lag  One might be interested in Model 𝐃𝐋(𝐪) using information on past interest rates to forecast future 1. Model representation: Lag values of inflation. the independent variable Can be used for policy analysis. 𝑌𝑡 = 𝛼0 + 𝛽0𝑋𝑡 + 𝛽1𝑋𝑡−1 + 𝛽2𝑋𝑡−2 + ⋯ + 𝛽𝑞𝑋𝑡−𝑞 + 𝜖𝑡 where 𝑡 = 𝑞 + 1, … , 𝑇  A central bank might be interested in how inflation will 2. Stationarity of variables and react now and in the future to a exogeneity change in the current interest 𝑌𝑡 and 𝑋𝑡 are stationary random rate. variables, and ∈𝑡 is independent of Distributed Lag Model 𝐃𝐋(𝐪) – current, past and future values of 𝑋, that Example is E(𝑋𝑡 , ∈𝑡 ) = 0  Suppose Autoregressive Model AR (p)  U denotes unemployment The value of a variable 𝑦 depends on rate the past values of itself, up to 𝑝 periods  G denotes GDP growth rate into the past.  We have 𝐃𝐋(𝐪) That is, That is, Distributed Lag Model 𝐃𝐋(𝟐) – Example Autoregressive Model 𝐀𝐑 (𝐩) − 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐢𝐞𝐫 Interpretation: (of impact (short-run) and total (long-run) multipliers) An impact multiplier of −𝟎. 𝟐𝟎𝟐 means that a 1-unit increase in GDP growth leads to 𝟎. 𝟐𝟎𝟐 decrease in the change in unemployment rate in the same period. A total multiplier of −𝟎. 𝟒𝟑𝟗 means that a 1-unit increase in GDP growth leads to 0.439 decrease in the change in unemployment rate in the long run. Distributed Lag Model 𝐃𝐋(𝟐) – Example Autoregressive Model 𝐀𝐑 𝐩 – Includes both autoregressive and Example distributed components in one model. Suppose That is,  G denotes GDP growth rate  We have 𝐀𝐑(𝐩) That is, Autoregressive Model 𝐀𝐑 𝟐 – Example Assumptions of the 𝐀𝐑𝐃𝐋(𝐩, 𝐪) Model Stationarity of variables 1. The variables are either stationary or cointegrated. It can handle both I(0) (stationary at level) and I(1) (non stationary but differenced to become stationary) variables. 2. ARDL models can be applied to datasets where variables are integrated of different orders (i.e., a mix of I(0) and I(1) variables), provided that they are not I(2) (second-differenced). Long-Run Relationship: 1. ARDL models are often used when there is an assumption of a long-run equilibrium relationship between the Autoregressive Distributed Lag dependent and independent variables Model ARDL(p,q) (i.e., cointegration). If such a relationship exists, the ARDL model can capture short-run and long-run dynamics. Possible Use of 𝐀𝐑𝐃𝐋 𝐩, 𝐪 Model Can be used for forecasting.  One might be interested to predict future unemployment rates based on historical economic growth rates and their past values. Can be used for policy analysis.  Governments might be interested to know the potential impact of different economic policies (e.g., fiscal stimulus or tax cuts) aimed at increasing economic growth to reduce unemployment rate. Establish economic relationships.  Researchers may analyze how changes in economic growth affect unemployment rate over both the short and long term.