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## Interpretation of log likelihoods

Discuss about the Commuters Cycling Behavior in Response to Weather Variation?

This report is about business cycle prediction in Sweden. Binary logistic regression method is used in this research paper for all the related statistical analysis. All the variables of interest are chosen with proper care and the significance of selecting each variable is analyzed and judged statistically. Researcher restricts the size of the test at alpha=0.05. In a binary business cycle operation, only 0 and 1 value are taken to predict outcomes. 0 value signifies an expansion in economic activities in a near future time period, 1 value codes the contraction in economic activity. The probabilities are derived from a number of variables that could have an impact on the real economy. Several related references are compiled at the reference list.

If the independent variables have a relationship to the dependent variables, the ability to predict the dependent variable accurately will improve, and the log likelihood measure will decrease thereof. The model fitted on quarterly data and the results of predicted and observed values are tabulated under recession and expansion outcomes. Expansion shows 82% and recession shows 75% correct prediction, and the overall correct percentage is obtained 78.9%.  Whereas in the case for monthly data, proportion of expansion and recession is 86% and 78.3% correct respectively. Over-all correct percentage is 82.6%. However, the log likelihood value for monthly data is 186.9, which is considerably large than that of quarterly data (68.2). Quarterly data model is thus, a better fit than monthly data model.

Note: -2 log likelihood is termed as badness of fit. The value below 100 indicates good fit and value under 20 ensures very good fit (Scott and Varian 2014).

In a binary classification, type I and type II errors are false alarms being positive and negative respectively. Not predicting a recession, which is likely to occur is type I error, and predicting the occurrence of false recession is type II error in this context. Almost 60% of total observations are correctly categorized in quarterly periodicity, and almost 56% of recessionary periods are properly categorized in prediction. The values for monthly periodicity are 76.5% and 75.9% respectively. Quarterly period has a proportional measure of 41% and the monthly period has the same measure of 21.2% in the context of the recessionary periods being false alarms.

Monthly data consists of a sample size, which is three times larger than that of quarterly data at the same time span. For any economic decision to predict recession or expansion, researcher needs to know at least the information of past 3 months in advance (Billio et al 2013).

The variables are used in both quarterly and monthly data models are the same but they differ in lag measures. The differences in selecting the variables in the sensitivity analysis are listed below in a table format.

 Europe GDP(3) Confidence (3) Oil(4) Europe GDP(8) OMX(1) Oil(12) Confidence(1) OMX(1) Spread(1) Spread(1) Building(1)

All the variables listed in the above table are statistically significant at 5% level. The values in the brackets indicate the time lag for each individual variable. Minimum three months lag is needed. Confidence (1), OMX(1), Spread(1)  variable and results in quarterly and building(1), OMX(1) and spread(1) variable for monthly are thus not so statistically reliable (Hamberg and Verständig 2009).

## Interpretation of classification tables

The Logit function for the quarterly data can be expressed as,

Lq(X) = 2.7- 115spread(1) – 4.2OMX(1) + 2.99Oil(4) – 55europeGDP(3) – 0.05confidence(1)

The logit function for the monthly data can be formulated as,

Lm(X) = 3.01- 103spread(1) – 4.3OMX(1) +2.8 Oil(12) – 65europeGDP(8) – 0.07confidence(3) – 1.73building(1).

OMX, spread, confidence, building and Europe GDP variables are expected to have a negative sign. That means, the associated variable if increases, there will be a lower probability of falling into economic recession. Similarly, it can be stated that higher value of the associated variable intrigues higher probability to fall into recession. The Oil variable is likely to have a positive sign. If the price of oil increases, the problem can be depicted as a supply shock and it can be responsible to affect economy negatively.

Analyzing the models researcher can claim that all OMX, spread, confidence, building and Europe GDP variables are associated with negative signed coefficients. In both the models, Oil variable is merged with positive valued coefficient. It can be concluded that both the models are fitted according to requirements and they are reliable (Tang and Shen 2014).

Probability of rejecting the null hypothesis is called the p value of the test. To find the validity of Beta coefficients two types of hypotheses are formulated, they are

Ho : beta equals to zero

H1 : beta not equals to zero.

Generally, the value of alpha is taken as 0.05. If the estimated p value is less than 0.05 or 5%, it provides enough confidence to reject the null hypothesis. In this analysis, not all the estimated p values are only significant at 5% level, they are so in the 10% level too. The null hypothesis that beta=0 can be rejected. In addition, the findings show beta values are not zero. It signifies the variables of interest in the particular study have some predictive powers. Largest beta coefficient suggests it has the greatest effect on the likelihood (Berge 2015).

Time lag has an important implication in the forecasting model. Generally, a lag of 3 is considered as statistically significant. According to the definition, a model fails to predict any further into the fore coming than its terse lag. In this report, the shortest lag implemented is one quarter and one month for the monthly data.  This suggests that the models should not be used to forecast any further events ahead than that lag. The Europe GDP figures here are lagged for two months. This fact confirms that a forecast implying a point of turning, which might actually appear at least three months earlier the actual GDP data has been published.

If the regression beta coefficient is positive, the interpretation is that for every 1-unit increase in the predictor variable, the dependent variable will increase by the non-standardized beta coefficient value.  Largest beta coefficient suggests it has the greatest effect on the likelihood. From the logit model described above, a logit function can be derived as,

P= eL(X)/ (1+eL(X))

Lower value of the logit function ensures less probability of falling into recession. The larger the beta coefficient is, it is likely that there will be more impact on recession. The coefficients with the OMX, spread, confidence, building and Europe GDP variables have negative sign. Proportion of recessionary periods correctly categorized in monthly data model is 75.9%, which is greater than that of quarterly. Monthly data model variables have thus greatest impacts and they are reliable to predict proper recession issue.

## Difference in variables between model

In this type of analysis researcher tries to predict which variables in both models are likely to depict recession or expansion of the economy. The Europe GDP variable has a negative beta coefficient. Thus, if there is any increase in GDP, it will reduce the likelihood of the recession. OMX, spread, confidence and building variables too have negative beta value, so increase in all these variables will shorten the likelihood of recession. The Oil variable has a positive beta coefficient. Thus, increase in the Oil variable will also increase the likelihood of a recession. The variables in both monthly and quarterly data model are fitted properly with the expected signs. Researcher can justify that the models confirm that the current economy under study, is expansionary (Ahmed, Rose, and Jakob 2016).

The time series predictors found in the in sample analysis are examined in contrast to the actual primary observations for the same time period. The integral model on monthly data explained above has a sample, which is three times larger as the one for the quarterly data model. By using this type of sample, monthly data model predicts almost 83% of the observations properly. In the out of sample prediction, the results from quarterly data can be viewed as weak. Comparatively frail out of sample forecast for quarterly model can be ascribed to the reduction of sample size.

Conclusion

As a conclusion based on the above discussions, researcher believes that the monthly data model consisting greater number of observations implements really well. The building variable is not significant enough with only (1) time lag. Other reason for the insignificance is building a house is a decision of long term and it should be planned in advance assorted years before the construction takes place in reality. Monthly data model ensures the findings in the main model excluding the building variable are indeed significant. The quarterly model is less sensitive to any changes in the dependent variable. A logistic regression model with the quarterly data set has multiple notable features. The results derived from the model should be considered as a guideline but not an absolute legitimacy (Scott and Varian 2014).

References

Ahmed, F., Rose, G. and Jakob, C., 2016. Commuters' Cycling Behavior in Response to Weather Variation: Insight from an Extended Theory of Planned Behavior. In Transportation Research Board 95th Annual Meeting (No. 16-5484).

Berge, T.J., 2015. Predicting recessions with leading indicators: model averaging and selection over the business cycle. Journal of Forecasting,34(6), pp.455-471.

Billio, M., Ferrara, L., Guegan, D. and Mazzi, G.L., 2013. Evaluation of Regime Switching Models for Realâ€ÂTime Business Cycle Analysis of the Euro Area. Journal of Forecasting, 32(7), pp.577-586.

Hamberg, U. and Verständig, D., 2009. Applying logistic regression models on business cycle prediction. Unpublished master's thesis, Stockholm School of Economics, Stockholm, Sweden). Retrieved from https://arc. hhs. se/download. aspx.

Scott, S.L. and Varian, H.R., 2014. Predicting the present with bayesian structural time series. International Journal of Mathematical Modelling and Numerical Optimisation, 5(1-2), pp.4-23.

Tang, J. and Shen, L.P., 2014. Application of Business Risk Prediction Model: Based on the Logistic Regression Model. International Journal of Business and Management, 9(7), p.139.

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