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Balance of Trade and Balance of Payments


 


Balance of Trade and Balance of Payments

Balance of Trade and Balance of Payments are two related terms but doesn’t mean the same. The main distinguishing factors are given below:

Important Formula for RBI Grade B DEPR Phase - 1 - Statistics and Econometrics

Karl Pearson's Correlation Coefficient

For Population:

\[{\rho _{XY}} = \frac{{cov(X,Y)}}{{{\sigma _X} \cdot {\sigma _Y}}}{\rm{ }}\]

For Sample:

\[\mathop r\nolimits_{XY} = \frac{{{s_{YX}}}}{{{s_X} \cdot {s_Y}}}\] \[\left\{ \begin{array}{l}Where,\\{s_{YX}} = \frac{{\sum {\left( {X - \overline X } \right)\left( {Y - \overline Y } \right)} }}{n}\\{s_X} = {\rm{ }}\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \\{s_Y} = \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} \end{array} \right.\]

Substituting the formula of Covariance and Standard Deviation: \[\mathop r\nolimits_{XY} = \frac{{\frac{{\sum {\left( {X - \overline X } \right)\left( {X - \overline X } \right)} }}{n}}}{{\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \cdot \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} }}\] Cancelling n: \[\mathop r\nolimits_{XY} = \frac{{\sum {\left( {{X_i} - \overline X } \right)\left( {{Y_i} - \overline Y } \right)} }}{{\sqrt {\sum {{{\left( {{X_i} - \overline X } \right)}^2} \cdot \sum {{{\left( {{Y_i} - \overline Y } \right)}^2}} } } }}\] Deviation Form: \[\mathop r\nolimits_{XY} = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sqrt {\sum {\mathop x\nolimits_i^2 } } \cdot \sqrt {\sum {\mathop y\nolimits_i^2 } } }}\] Formula without Mean and Deviation: Formula without Mean and Deviation: \[\mathop r\nolimits_{XY} = \frac{{n\sum {\left( {{X_i}{Y_i}} \right) - \left( {\sum {{X_i}} } \right)\left( {\sum {{Y_i}} } \right)} }}{{\sqrt {n\sum {X_i^2 - {{\left( {\sum {{X_i}} } \right)}^2}} } \cdot \sqrt {n\sum {Y_i^2 - {{\left( {\sum {{Y_i}} } \right)}^2}} } }}\]

Spearman's Rank Correlation Coefficient:

\[r' = 1 - \frac{{6\sum {{D^2}} }}{{n\left( {{n^2} - 1} \right)}}\] Where,
D = Difference between the ranks of corresponding pairs of X and Y

Simple Linear Regession Model:

For Population

\[\begin{array}{l}{Y_i} = {b_0} + {b_1}{X_i}\\{Y_i} = {b_0} + {b_1}{X_i} + u\end{array}\]

For Sample

\[{{\hat Y}_i} = {{\hat b}_0} + {{\hat b}_1}{X_i}\]

OLS Estimation

\[\begin{array}{l}{Y_i} - {{\hat Y}_i} = {e_i}\\Minimize \to {e_i} = {Y_i} - {{\hat Y}_i}\\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat Y}_i}} \right)}\nolimits^2 } \\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat b}_0} + {{\hat b}_1}{X_i}} \right)}\nolimits^2 } \\Slope:\\\mathop {\hat b}\nolimits_1 = \frac{{n\sum {\mathop X\nolimits_i \mathop Y\nolimits_i - \sum {\mathop X\nolimits_i \sum {\mathop Y\nolimits_i } } } }}{{n\sum {\mathop X\nolimits_i^2 - \mathop {\left( {\sum {\mathop X\nolimits_i } } \right)}\nolimits^2 } }}\\\mathop {\hat b}\nolimits_1 = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sum {\mathop x\nolimits_i^2 } }} = \frac{{{\mathop{\rm cov}} (X,Y)}}{{\mathop \sigma \nolimits_X^2 }}\\Intercept:\\\mathop {\hat b}\nolimits_0 = \overline Y - {{\hat b}_1}\overline X \end{array}\]

Hypothesis Testing of estimators:

Null Hypothesis = Estimators are not statistically more than zero.
\[\begin{array}{l}\mathop t\nolimits_0 = \frac{{{{\hat b}_0} - {b_0}}}{{{s_{{{\widehat b}_0}}}}}\\\mathop t\nolimits_1 = \frac{{{{\hat b}_1} - {b_1}}}{{{s_{{{\widehat b}_1}}}}}\end{array}\]

Variance of estimators when population variance is given:

\[\begin{array}{l}Var{\rm{ }}{\widehat b_0} = \sigma _u^2\frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\Var{\rm{ }}{\widehat b_1} = \sigma _u^2\frac{1}{{\sum {x_i^2} }}\end{array}\]

Estimator of Population Variance (\(\sigma _u^2\)):

\[{s^2} = \hat \sigma _u^2 = \frac{{\sum {e_i^2} }}{{n - k}}\]

Variance of estimators when population variance is not given:

\[\begin{array}{l}s_{{{\widehat b}_0}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\s_{{{\widehat b}_1}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{1}{{\sum {x_i^2} }}\end{array}\]

Coefficient of Determination:

\[\mathop R\nolimits^2 = \frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }} = 1 - \frac{{\sum {e_i^2} }}{{\sum {y_i^2} }} = \mathop {\hat b}\nolimits_1 \frac{{\sum {{x_i}{y_i}} }}{{\sum {y_i^2} }}\]

Relationship between Correlation Coefficient & Coefficient of Determination:

\[r = \sqrt {{R^2}} \]

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