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.
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.
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.
Data is vital since it is used to evaluate five critical issues used in the overall estimation of the proposed research.
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.
When there is some slight missing data hence calling for assumptions to be made.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.