**Time series**. It is the case when the impact of an independent variable is expected to range out over some periods.

**Stationary series time** has constant mean X(t) over time, its variance X(t) is constant over time, and its simple correlation coefficient between X(t) and X(t-k) is dependent on the length of the lag(k) but not on any other variable (for all k) involved. While the nonstationary time series X(t) is formed if one or more of the properties are not met.

**Distributed lag model**. It explains the current value of Y as a function of current and previous values of X.

**Simple lags**. They are independent variables that can be used if X is expected to impact Y after a certain period.

**Dynamic models.** It is used to represent a distributed lag model.

**Bias in Dynamic models**. It is the effect of serial correlation in dynamic models, hence inconsistency.

**Spurious correlation** is a strong relationship between more than two variables while its cause is not a real underlying instrumental relationship. It can be caused by non-stationarity.

**Binomial logit model.** It is an estimation practice that escapes the unboundedness problem by using a variant of the cumulative function.

**Nature and examples** of the Y of simultaneous equation systems clearly affect at least one X(s), but the Y is not affected by X(s). They are majorly found in the economic world.

**Econometric issues with simultaneous systems.**

The variables involved in the simultaneous equation are typical econometric equations with regression and linear relationship. An effect on one of the variables affects the other.

**Two-stage least squares.** It offers as an alternative to mitigate simultaneity bias. It is done under two steps; to find highly correlated with the independent variable and then uncorrelated with the error term.

**Reduced form equations.**

They express a particular independent variable in terms of an error term and all prearranged variables present in the simultaneous method.

**Impact multipliers. **

Reasons to use reduced form equations in estimations with simultaneous equations

**Examples of endogenous, exogenous variables.** Endogenous variables; supply and demand of the soft-drink industry. Exogenous variables; supply and demand synchronous system.

**Simultaneous bias.** It states to the element that in a real-time system, OLS-estimated coefficients are biased.

**Identification condition. **

It states that a structural system has identification if and only if the enough of the system’s prearranged variables are mislaid from the equation to be eminent from all the others in the order.

**The accuracy of forecasting.** It is the accurate estimating of independent variables.

**The confidence interval of a forecast.** It is a collection of values holding the actual value of the item estimation at a specified percentage of the time.

**ARIMA models.** These are a highly refined curve-fitting device using both the present and past

Values of the dependent variables to yield short-range predictions of that variable.

**Comment critically: “Data are the most essential component of the empirical analysis. It should be complete.**

Data is vital since it is used to evaluate five critical issues used in the overall estimation of the proposed research.

**Present it in the conventional (standard) way with its auxiliary information. Speculate on the possible issues with your results.**

Y(35)=α(0) + β(0)X(1) + β(2)X(2) + β(3)X(3) + ε. The data might be highly biased and accurate.

Betas are the coefficients of the explanatory variables.

** **

**In a typical regression study, explore occasions (imagine situations) when the researcher must resort to gut feelings or live with the second best.**

When there is some slight missing data hence calling for assumptions to be made.

** **

**Explain a typical dynamic lag equation, issues with it and possible solutions.**

** **

It explains the current value of Y as a function of current and past values of X. various lags of X have a high probability of having severe multicollinearity. It is also not guaranteed that the estimated betas will track the smoothly declining pattern proposed by the theory. Serial correlation causes a bias for the models. Use of OLS to tackle serial correlation especially in small samples.

**How is the LM multiplier performed in this case?**- You are obtaining the residuals from an estimated equation.
- Specification of the auxiliary equation.

- Use of OLS in the estimation of auxiliary and testing of the null hypothesis that [a(3)=0].

**If a researcher is considering whether to use distributed lags or proceed with a dynamic model, discuss the considerations to use one or the other.**

When using a dynamic model can be used to represent a distributed model and simple. The coefficients of the lagged X’s follow a clear pattern.

**Estimation methods of models with dummy variables.**

Linear probability model where it is a linear-in-the-variables equation used to explicate a dummy and dependent variable (Chamroukhi, p. 640). Binomial model, an estimation technique which avoids the unboundedness problem by using a variant of the cumulative normal distribution.

**Possible issues with a binary-dependent-variable model.**

The model can have an outcome with two possibilities; a success or a failure. The dependent variable Y is interpreted as a modeling, with the probability of one.

**Provide your own example of a complete model with a binary-dependent variable**

Dependent variable explaining if a company j (j=1, 2, … , N) has a van (y(i)=1) or not (y(i)=0).

Let the van ownership to be a function of company income, with the exogenous variable x(i).

The linear regression model of van ownership is:

y(i) = βx(i) + µ(i), i=1, 2, …n

Independent variables are

**Explain estimation methods for simultaneous equation systems.**

Reduced-form equations. It expresses the endogenous variable in terms of both the error term and all predetermined variables, (Cai, et al., p. 63) in the simultaneous system.

Two-Stage least squares estimation method where it estimates the coefficients of a simultaneous system.

**Prices and consumption model**

Pr(t) = µ + β(0)CO(t) + β(1)CO(t-1) + ε(t)

Where Pr is the price of commodities and CO is the consumption rate of the people.

Pr(t) = -266.6 + o.45CO(t) + 0.28CO(t-1)

t= 4.71 5.67

R^2= 0.987 N= 32

Approximation can be done by inserting the required variables into the model and obtain the results.

**Discuss the use and issues with ARIMA models.**

It is used to yield interim forecasts of dependent variables. They ignore the independent variables and any underlying theories. It is apt when little is identified about the dependent variable. They can beat moderately sophisticated econometric models.

**Discuss the factors that contribute to the reliability of a forecast.**

A careful selection of independent variables, which aids to avoid the need for conditional forecasting. Specification and estimation of the equation with the dependent variable being the item to be forecasted.

**References**

Cai, Qisen, et al. “A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression.” *Knowledge-Based Systems* 74 (2015): 61-68.

Chamroukhi, Faicel, et al. “Joint segmentation of multivariate time series with hidden process regression for human activity recognition.” *Neurocomputing* 120 (2013): 633-644.