Problem Description:
The data analysis assignment using SPSS focuses on regression analysis and forecasting using different models. The first part, involves computing standardized values, predicting test scores, and exploring the limitations of Ordinary Least Squares (OLS) in Big Data. The second part delves into time series analysis, specifically ARIMA and vector autoregression models, to forecast and analyze unemployment rates. The final section discusses the impact of real oil price changes on GDP growth and involves statistical hypothesis testing and the estimation of cumulative dynamic multipliers.
Regression Analysis:
- Standardized Values and Prediction
Standardized RPM: -0.29
Standardized RPM = (0.52 - 0.6)/0.28 = -0.29
Standardized TEXP: -0.55
Standardized TEXP = (11.1 - 13.2)/3.8 = -0.55
Predicted TestScore: 759.77
β1 = Cov(RPM,TestScore)/Var(RPM)
β2 = Cov(TEXP,TestScore)/Var(TEXP)
α = mean(TestScore) - β1 x mean(RPM) - β2 x mean(TEXP)
Prediction Error: 15.53
- Regression Coefficients
Regression Equation:
Cov(RPM,TestScore) = -19.924,Var(RPM) = 0.0784
Cov(TEXP,TestScore)= 92.722,Var(TEXP) = 14.44
β1 = - 19.924/0.0784 = -254.29
β2 = 92.722/14.44 = 6.41
α = 750.1 - (-254.29 x 0.6) - (6.41 x 13.2) = 918.5
Therefore, the regression equation is:
(TestScore) ̂ = 918.5 - 254.29 x RPM + 6.41 x TEXP
- Limitation of OLS in Big Data
Solution: Use regularized regression methods (Lasso, Ridge, Elastic Net) for high-dimensional data.
Time Series Analysis:
- Create a variable called dunrate that is calculated by differencing unrate variable
- Compute an AR(1) model for dunrate series for the period 1960m1 to 2023m2
AR(1) Model Equation:
ARIMA regression
Sample: 1960-01 - 2023-02 Number of obs = 758
Log likelihood = -443.207 Prob> chi2 = 0.0021
------------------------------------------------------------------------------
| OPG
dunrate | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dunrate |
_cons | -.0022754 .0310869 -0.07 0.942 -.0632046 .0586538
-------------+----------------------------------------------------------------
ARMA |
ar |
L1. | .0308947 .0100318 3.08 0.002 .0112328 .0505566
-------------+----------------------------------------------------------------
/sigma | .4342182 .0014395 301.64 0.000 .4313968 .4370397
Table 1: AR(1) Model equation
Forecast (March 2023):To be calculated
- Write your model in part (b) as a regression equation
The regression equation for the AR(1) model will be of the form:
dunrate_t = -0.002 + 0.031*dunrate_(t-1)
- Vector Autoregression Model
ADL(1,1) Model Equation: dunrate_t = -0.041 - 0.062 * dunrate_(t-1) + 0.991 * inf_(t-1)
Forecast (March 2023):To be calculated
- Forecast Comparison
Result:The forecast from the ADL(1,1) model is closer to real data than the AR(1) model.
Compute an ADL(1,1) model for the dunrate series adding a lagged inflation rate to your model for the period 1960m1 to 2023m2
Vector autoregression
Sample: 1960-01 - 2023-02 No. of obs = 757
Log likelihood = -779.8475 AIC = 2.076215
FPE = .0273363 HQIC = 2.090348
Det(Sigma_ml) = .0269064 SBIC = 2.112908
Equation Parms RMSE R-sq chi2 P>chi2
----------------------------------------------------------------
dunrate 3 .434087 0.0057 4.319347 0.1154
inf 3 .384332 0.9818 40850.98 0.0000
----------------------------------------------------------------
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dunrate |
dunrate |
L1. | .0274544 .0362668 0.76 0.449 -.0436271 .098536
|
inf |
L1. | .0105087 .0055418 1.90 0.058 -.000353 .0213705
|
_cons | -.0413398 .0261912 -1.58 0.114 -.0926736 .0099939
-------------+----------------------------------------------------------------
inf |
dunrate |
L1. | -.0622813 .0321099 -1.94 0.052 -.1252155 .0006529
|inf |
L1. | .9909436 .0049066 201.96 0.000 .9813268 1.00056
|
_cons | .0388204 .0231892 1.67 0.094 -.0066295 .0842703
------------------------------------------------------------------------------
Table 2: ADL (1,1) for Dunrate series
Oil Price and GDP Growth:
- Impact of 25-Percentage Point Increase in Real Oil Prices
Impact Effect: -0.225 (decrease in GDP growth)
- Predicted Cumulative Change in GDP Growth
Change: -ΔY_t = Y_t - Y_(t_(-2) )
ΔY_t = -0.009ChangRPoil_t - 0.028ChangRPoil_(t-1)
ΔY_t = -0.009 * 25 - 0.028* 25 = -0.885
- Hypothesis Testing
HAC F-Statistic: 4.07
Conclusion: Reject null hypothesis; oil price changes have a significant effect on real GDP growth.
- Estimation for Cumulative Dynamic Multipliers
Equation:
ΔY_t= β_0 + β_1 ChangRPoil_t + β_2 ChangRPoil_(t-1) + ...+ β_k ChangRPoil_(t-k) + u_t
Process: Calculate cumulative dynamic multipliers and standard errors from estimated coefficients.