DEFINING KEY CONCEPTS IN THE QUESTION PAPER
MODELLING LONG RUN RELATIONSHIPS
Stationarity and unit root testing
- A stationary series can be defined as one with a constant mean, constant variance, and constant autocovariance. The use of non-stationary data can lead to spurious regression – normally very high R- squared greater the Durbin Watson
Testing for unit root
- ADF
- Philips Perron
Cointegration
- Cointegration is an important tool for modeling the long-run relationships in time series data. Economic theory suggests that many times series variables move together in the long run or fluctuating around a long run equilibrium and any divergence between variables is a short run Cointegration occurs when two or more non-stationary time series:
- Have a long-run equilibrium
- Move together such that their linear combination results in a stationary series
- Share underlying stochastic trend
Test for cointegration
- Residual based test
- Johansen technique
Error Correction Model
- Cointegration implies that the time series will be connecting through an error correction model. The error correction model:
- Reflects long-run relationships of variables
- Includes short-run dynamic adjustment mechanism that describes how variables adjust when they are out of equilibrium
LIMITED DEPENDENT VARIABLE MODELS
Linear probability models
There are many situations in research where the dependent variable is qualitative. The qualitative information will be coded as a dummy variable and the situation would be referred a limited dependent variable. In our case in question 2, the dependent variable is binary where 1 is accepted into an honors module and 0 is not accepted into an honors module. The use of OLS is not perfect when estimation models with a limited dependent variable.
Logit and profit approaches
These are approaches used to overcome the limitation of the LPM that it can produce probabilities that are negative and greater than 1. They do this by using a function that transforms the regression model so that fitted values are bounded between 0 and 1 interval.
1a) Use the ADF test to test all four variables for unit roots. Provide your answers in the table below (Hint: please remember to log variables before performing the tests):
| Variable | Model | Lags | ADF test statistic | Prob | Interpretation |
|
LNS |
Trend and Intercept | AIC 1 | -2.080231 | 0.5418 | Non-Stationary |
| Intercept | AIC 1 | -2.080231 | 0.7841 | Non-Stationary | |
| None | AIC 1 | -2.080231 | 0.9526 | Non-Stationary | |
|
DLNS |
Trend and Intercept | AIC 1 | -3.457736 | 0.0571 | Non-Stationary |
| Intercept | AIC 1 | -3.487169 | 0.0131 | Stationary | |
| None | AIC 1 | -3.487169 | 0.0023 | Stationary | |
|
LGDP |
Trend and Intercept | AIC 1 | -0.025106 | 0.6983 | Non-stationary |
| Intercept | AIC 1 | -1.777274 | 0.9508 | Non-stationary | |
| None | AIC 1 | 3.384360 | 0.9997 | Non-stationary | |
|
DLGDP |
Trend and Intercept | AIC 1 | -4.627472 | 0.00031 | Stationary |
| Intercept | AIC 1 | -4.671391 | 0.0005 | Stationary | |
| None | AIC 1 | -4.671391 | 0.0049 | Stationary | |
|
LLC |
Trend and Intercept | AIC 1 | -1.217220 | 0.8942 | Non-Stationary |
| Intercept | AIC 1 | -1.876704 | 0.3398 | Non-Stationary | |
| None | AIC 1 | -1.876704 | 0.7859 | Non-Stationary | |
|
DLLC |
Trend and Intercept | AIC 1 | -4.817874 | 0.0018 | Stationary |
| Intercept | AIC 1 | -4.396242 | 0.0011 | Stationary | |
| None | AIC 1 | -0.778159 | 0.3728 | Non-Stationary | |
|
LLP |
Trend and Intercept | AIC 1 | -2.665692 | 0.2554 | Non-Stationary |
| Intercept | AIC 1 | -1.391756 | 05767 | Non-Stationary | |
| None | AIC 1 | -1.391756 | 0.8465 | Non-Stationary | |
|
DLLP |
Trend and Intercept | AIC 1 | -2.887812 | 0.1770 | Non-Stationary |
| Intercept | AIC 1 | -2.922536 | 0.0516 | Non-Stationary | |
| None | AIC 1 | -2.869805 | 0.0052 | Stationary |
1 b) Test for cointegration between variables:
Date: 01/11/21 Time: 07:58
Sample (adjusted): 1973 2014
Included observations: 42 after adjustments
Trend assumption: Linear deterministic trend
Series: LNS LGDP LLC
Lags interval (in first differences): 1 to 2
Unrestricted Cointegration Rank Test (Trace)
| Hypothesized No. of CE(s) |
Eigenvalue |
Trace Statistic | 0.05
Critical Value |
Prob.** |
| None | 0.302209 | 24.18886 | 29.79707 | 0.1926 |
| At most 1 | 0.147577 | 9.075745 | 15.49471 | 0.3584 |
| At most 2 | 0.054854 | 2.369482 | 3.841466 | 0.1237 |
Trace test indicates no cointegration at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
| Hypothesized No. of CE(s) |
Eigenvalue |
Max-Eigen Statistic | 0.05
Critical Value |
Prob.** |
| None | 0.302209 | 15.11311 | 21.13162 | 0.2811 |
| At most 1 | 0.147577 | 6.706263 | 14.26460 | 0.5244 |
| At most 2 | 0.054854 | 2.369482 | 3.841466 | 0.1237 |
Max-eigenvalue test indicates no cointegration at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegrating Coefficients (normalized by b’*S11*b=I):
(i) Estimate the following long-run cointegration equation and use your results to complete the table. (Remember to include an intercept term.
LNS = f (LGDP, LLC)
| Dependent Variable: LNS | ||||
| Method: Least Squares | ||||
| Date: 01/10/21 Time: 23:11 | ||||
| Sample: 1970 2014 | ||||
| Included observations: 45 | ||||
| Variable | Coefficient | Std. Error | t-Statistic | Prob. |
| LGDP | 1.298990 | 0.163023 | 7.968145 | 0.0000 |
| LLC | -0.129727 | 0.034002 | -3.815242 | 0.0004 |
| C | -14.06660 | 2.252832 | -6.243961 | 0.0000 |
| R-squared | 0.881356 | Mean dependent var | 4.241641 | |
| Adjusted R-squared | 0.875706 | S.D. dependent var | 0.233948 | |
| S.E. of regression | 0.082479 | Akaike info criterion | -2.088202 | |
| Sum squared resid | 0.285718 | Schwarz criterion | -1.967757 | |
| Log-likelihood | 49.98453 | Hannan-Quinn criter. | -2.043301 | |
| F-statistic | 156.0003 | Durbin-Watson stat | 0.284335 | |
| Prob(F-statistic) | 0.000000 | |||
ii) Interpretation of coefficients
There is a positive relationship between GDP and demand for skilled labor, meaning that as the country expands economically the demand for skilled labor increases. A percentage change in GDP will result in a 1.298% increase in demand for skilled labor all things being equal.
There is a negative relationship between labor costs and demand for skilled labor, meaning that as labor costs increase demand for skilled labor decreases. A percentage change in labor costs will result in a 0.1297% decrease in demand for skilled labor.
iii) Yes, the coefficients correspond to priori expectations because theoretically GDP is positively related to the demand for skilled labor. Also, costs are negatively related to the demand for skilled labor both in theory and practice.
ii) Generate residual series
| Null Hypothesis: RESID02 has a unit root | |||
| Exogenous: Constant | |||
| Lag Length: 1 (Automatic – based on SIC, maxlag=9) | |||
|
|
t-Statistic | Prob.* | |
| Augmented Dickey-Fuller test statistic | -2.776454 | 0.0701 | |
| Test critical values: | 1% level | -3.592462 | |
| 5% level | -2.931404 | ||
| 10% level | -2.603944 | ||
| *MacKinnon (1996) one-sided p-values. | |||
At 1% and 5% levels of significance, the residual series has a unit root meaning nonstationary, however stationary at 10% level of significance the series is stationary.
- Since residual series are not stationary at a 5% level of significance we can conclude that there is no cointegration between variables. The results are in line with the cointegration test
C) Build an Error Correction Model (ECM) for the demand for skilled labor
| Dependent Variable: D(LNS) | ||||
| Method: Least Squares | ||||
| Date: 01/10/21 Time: 23:17 | ||||
| Sample (adjusted): 1971 2014 | ||||
| Included observations: 44 after adjustments | ||||
| Variable | Coefficient | Std. Error | t-Statistic | Prob. |
| D(LGDP) | 0.892320 | 0.053556 | 16.66151 | 0.0000 |
| D(LLP) | -0.970114 | 0.032150 | -30.17447 | 0.0000 |
| RESID02(-1) | -0.036667 | 0.015163 | -2.418204 | 0.0202 |
| C | 0.002268 | 0.001746 | 1.298846 | 0.2014 |
| R-squared | 0.971835 | Mean dependent var | 0.019602 | |
| Adjusted R-squared | 0.969722 | S.D. dependent var | 0.043222 | |
| S.E. of regression | 0.007521 | Akaike info criterion | -6.855758 | |
| Sum squared resid | 0.002263 | Schwarz criterion | -6.693559 | |
| Log-likelihood | 154.8267 | Hannan-Quinn criter. | -6.795607 | |
| F-statistic | 460.0599 | Durbin-Watson stat | 2.891087 | |
| Prob(F-statistic) | 0.000000 | |||
ii) The error correction term should be negative, significant, and less than a unit. In our case, the error correction term is negative, less than 1, and statistically significant and 5% level of significance.
d) Perform diagnostic checks on the ECM
| Test | Null Hypothesis | Test statistic | P-value | Conclusion |
|
Jarque-Bera |
H𝑜𝑜: Normally distributed residuals |
JB = 27.50960 |
0.0000001 |
We reject Ho. Residuals are not normally
distributed |
|
Ljung –Box Q |
H𝑜𝑜: No serial correlation |
LBQ(6) = 11.954 |
0.063 |
We failed to reject Ho at a 5% level of significance.
Residuals are not serially correlated |
| Breusch- Godfrey LM TEST | H𝑜𝑜: No serial correlation |
𝑛𝑛R2(2) = 15.44363 |
0.0004 |
We reject Ho. Residuals are
serially correlated |
|
ARCH-LM |
H𝑜𝑜: No hetroscedasticity |
𝑛𝑛R2(2) =9.327868 |
0.0094 |
We reject Ho. There is a presence of
Heteroscedasticity |
|
White |
H𝑜𝑜: No hetroscedasticity |
𝑛𝑛R2(𝑛𝑛𝑛𝑛 𝐶𝐶𝐶𝐶) = 9.327868 |
0.0000 |
We reject Ho. There is a presence of
Heteroscedasticity |
|
Ramsey RESET |
H𝑜𝑜: No misspecification |
LR (2) = 2.861188 |
0.2392 |
We failed to reject Ho. No
misspecification |
ii) Given your conclusions on the diagnostic check of the ECM, do you think that this is an acceptable model
The model is not acceptable because it has violated some of the OLS assumptions. The model suffers from the problem of serial autocorrelation therefore estimates won’t be BLUE (Best Linear Unbiased Estimators), and they won’t be reliable enough. The model suffers from the problem of Heteroscedasticity. If errors are heteroscedastic it will be difficult to trust the standard errors of the OLS estimates. Hence, the confidence intervals will be either too narrow or too wide. This impact will forecasting and variance decomposition.
e) Regardless of the results you obtained in question 1(d), suppose you still decide to create a model statement in EViews to combine your long run and ECM.
The purpose of this step is to re-write the equation back to its levels and simulate it dynamically. The outcome is to create a new modeled variable of the dependent variable.
Step 1
Long run
𝑦𝑦𝑡𝑡 = 𝑦𝑦^𝑡𝑡 + 𝑢𝑢^𝑡𝑡 – The estimated cointegrating equation We rewrite the equation so the residuals are on the left
𝑢𝑢𝑡𝑡 = 𝑦𝑦𝑡𝑡 + 𝑦𝑦^𝑡𝑡
Step 2
ECM
We specify the ECM model so that the differenced dependent variable 𝐷𝐷LN𝐷𝐷𝑡𝑡 is the dependent variable in the equation. The purpose is to rewrite the equation back to levels.
By differencing we mean
𝑑𝑑(𝑦𝑦𝑡𝑡) = 𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑡𝑡−1
𝑑𝑑(LN𝐷𝐷𝑡𝑡) = LN𝐷𝐷𝑡𝑡 − LN𝐷𝐷𝑡𝑡−1
Re-write the ECM so the LNS becomes a new dependent variable and the rest of the equation stays the same. We simply add LN𝐷𝐷𝑡𝑡−1 at the end
Step 3
Get rid of the logs NS = exp (logNS)
- Provide the missing values/variables in the model statement (please write your answer next to the correct option in the space provided below the statement):
LNS = 1.2989899682*LGDP – 0.129726760054*LLC – 14.0665958153 – Long run model
D (LNS) = 0.892320312733*D (LGDP) – 0.970114223165*D (LLP) – 0.0366672785059*RESID02 (-1) + 0.00226826179642 – ECM Model
RELNS = LNS – 1.2989899682*LGDP – 0.129726760054*LLC – 14.0665958153
D (LNS) = 0.892320312733*D (LGDP) – 0.970114223165*D (LLP) – 0.0366672785059*RESID02 (-1) + 0.00226826179642 + LNS (-)
NS = EXP(LNS)
i) Graph the actual and estimated values for the demand for labor. Comment on the fit you observe. (Hint: copy/paste your graph of LNS and LNS (Baseline).)