# Correlation between productivity and gdp in the uk

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The numbers of Gross domestic product (GDP) are generally used to evaluate the overall performance of a country's economy. To take a closer look, it shows that productivity enhancements are the primary source for supporting economic growth (Mckinsey, 2002). Productivity is generally defined as output per unit of input and is an essential indicator of using resources efficiently (Phelps, 2010). In the long-term, it also is a signal of underlying economic growth potential. It reflects the efficiency of labour market which interacts with the changes in production and output (Myers, 2009). Furthermore, a fall in GDP growth will unavoidably have an incidental impact on growth of productivity. The timing of reflection the changes depends on the situation in the labour market. Although GDP growth can drop quite rapidly, and sometimes suddenly, companies tend to delay the action of lay off workers or decrease hour worked, namely, the action of diminishing the amount of labour employed behinds of a drop GDP growth. The hour worked are likely to be reduced more quickly than amount of hiring staff, as a result, there is a lag compared with the change of GDP growth (Myers, 2009)

There are two factors influence the per capita of GDP which are employment levels and labour productivity. Employment levels refer to the ratio of the actively labour force involved in economic activities and the average hour worked. In the other hand, labour productivity is the output produced by per unit of labour. To sustain the growth of GDP per capita, labour productivity must increase. For example, when a company improves its output per unit of labour that allows this company to obtain more profits, and have the ability to pay higher salaries to employees. This surplus will be directed back to the economy by increased consumer expenditure, higher exports, and more business investment. Consequently, the increasing in productivity leads to GDP increased (Mckinsey, 2002).

Both the information of Gross Value Added (GVA) per hour worked and per worker measures are used to compare GDP growth with productivity growth. From 25 years data, it indicates that productivity growth is very little lag behind to respond the changes in GDP growth (Myers, 2009). From this finding, it shows a relatively sensitive and flexible between labout market and the changes in output growth. For example, when the labour market is rigid and output ran down, it is difficult to hire properly skilled and experienced staff and to reduce work hours or lay off staff. Also, the growth rate of productivity tends to decline in front of the fall in output growth in economy downturn, but increase ahead of rise in output growth when economy recovers (Myers, 2009).

The trend of growth implies the underlying growth potential of the economy that outside the cyclical influences. It is a main determinant of the long-term growth rate of an economy, and acts as the base in deciding the growth of per capita income and the direction of living standards. Although short term causes may influence the growth of GDP and the living standards, in the long-term, it underpins the potential growth of the economy that resolve the wealth and prosperity of a country and living standard. To estimate the trend growth, it is based on the average growth rate between the start and the end of economic cycle, the productivity growth, average hour worked, the employment rate and the adult population are also included (Myers, 2009).

The Higher productivity not only directly enriches national welfare; it also enhances the competitive abilities of companies and national economies, allowing growth without inflation, and creating a financial platform for social spending. Eventually, productivity is the only sustainable factor for job creation (Mckinsey, 2002).

## Data description and collection

In this paper, it intends to explore the correlation and regression between productivity and GDP in UK. The two variables come from different database but from the same organization which is Office for National Statistics (ONS). The data of 'Manufacturing output per filled job, unit wage costs and productivity jobs' is related to the UK manufacturing sector. The output per filled job is the proportion of Index of Production and productivity jobs in manufacturing industry. The major area for the data collection is at UK. This data is released monthly and quarterly by ONS. The quarterly data for manufacturing could trace back to Q4 1954 (Great Britain. Office for National Statistics, 2009).

The Office for National Statistics (ONS) preliminary estimates for gross domestic product (GDP) including steady price gross value added (GVA) data of UK. Data are separated by division of industry. The main region of GDP data is collected from UK and the GDP data is released by quarterly. In this paper, the GDP data will be used from Q1 1985 to Q4 2009. The quarterly GDP growth rates of last quarter of previous year are used to analyze the quarterly productivity growth rate of last quarter of previous year in UK, finding out the relationship with these two variables. These two statistics are produced by following high professional principles and announced base on the arrangements permitted by the UK Statistics Authority (Great Britain. Office for National Statistics, 2010).

## Analysis

## Scatter diagrams pattern

To analyze the relationship between two quantitative variables, the scatter plot is used to graphically represent the relationship between two variables (Groebner et al, 2008). The scatter diagrams help to examine the data graphically and draw the initial conclusion about the possible relationship between variables. On Figure 1, the scatter diagrams show the positive relationship between quarterly GDP growth and quarterly productivity growth. It should be possible to estimate productivity growth by given GDP growth forecasts.

Figure 1: Scattergrams

## 3.2 Correlation

## Correlation coefficient

The correlation coefficient determines the relative strength of a linear relationship between two numerical variables. The values of the correlation coefficient range from -1 for a perfect negative correlation to +1 for a perfect positive correlation. The value of correlation coefficient is closer to +1 or -1, and then the linear relationship between the two variables is stronger. The exiting of a strong correlation does not mean there is a causation effect. It only shows the tendencies presented in the data (Berenson et al, 2009).

It shows that quarterly GDP growth has the relation with quarterly productivity growth. Also, r=0.7394 indicates the two variables have strong relation.

## Confidence intervals for the correlation coefficient

FISHER

w

Correlation coefficient (r)

0.7394

0.9492

sample size (n)

100

standard deviation

0.1015

95% confidence interval

from

0.6353

0.7503

To

0.8171

1.1481

Figure 2: Confidence intervals for the correlation coefficient

Based on sample data to make an inference about a population value of the correlation coefficient, The Fisher's transformation is used. It is a formula which firstly transform the value r to a value of w, that is approximately normally distributed with variance (n is the sample size). And finally, transform back to values for r (Jessop, 2010).

The mean for the normal distribution:

0.9492

The standard deviation is ==0.1015

The 95% confidence interval for w is

0.7503 to 1.1481

Then use FISHERINV to get the corresponding values of r:

w = 0.7503 ® r = FISHERINV (0.7503) = 0.6353

and w = 1.1481 ® r = FISHERINV (1.1481) = 0.8171

The 95% confidence interval for the population correlation is from 0.6453 to 0.8171.

## 3.3 Regression model

## Simple linear regression model

There are two parameters which are slope and intercept composed the linear model. A linear model has the form y = a + bx which describes how the mean value of y is related to x.

a is the intercept, the value of y when x=0.

b is the slope, the change in y when x increases by 1.

To present linear regression, it is y=intercept + (slope * x) = a + bx

a= intercept

-0.1546

b = slope

0.1365

y = -0.1546 + 0.1365 x

Figure 3: Linear regression

The slope is 0.1365; it means that for each increase of 1 unit in x, the mean value of y is estimated to increase by 0.1365 units. It could be assumed that if quarterly GDP growth rate increases 5% then estimate the productivity growth will be 0.5279.

## inference: regression analysis

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.7394

R Square

0.546712

Adjusted R Square

0.542087

Standard Error

0.270283

Observations

100

## ã€€

Coefficients

Standard Error

T Stat

P-value

Lower 95%

Upper 95%

Intercept

-0.15458

0.040204

-3.84488

0.000214

-0.23436

-0.0748

X

0.136474

0.012553

10.8719

1.57E-18

0.111563

0.161385

Figure 4: Summary output

The overall model performance of regression statistics is showed on Figure 4. The Multiple R represents correlation coefficient of two variables which are quarterly GDP growth and quarterly productivity growth and the number is 0.7394. The amount of observations, which is the sample size, is 100, checking the table of 5% risk of incorrectly concluding non-zero population correlation, the number is 0.197. The value of sample CORREL, which is 0.7394, is larger than 0.197. It concludes that in the population CORREL is non-zero correlation. That is to say, there is a relation between the two variables (Jessop, 2010).

R square is the coefficient of determination in regression analysis. It means the ratio of the total variation in the dependent variable can be explained by its relative independent variable (Groebner et al, 2008). R Square is 0.5467. It means 54.67% of the variation in the quarterly productivity growth data for this sample can be explained by the linear relationship between quarterly productivity growth and quarterly GDP growth (Groebner et al, 2008).

Adjusted R Square specifies the sample size and is a better way to estimate the population. In a simple regression analysis, adjusted (Jessop, 2010).

Standard Error is the standard deviation of the residuals and is a compute of the level of variation in variable y remains cannot be accounted by the model (Jessop, 2010). An inference of the population standard deviation of residuals is

Standard error

That = 0.2703 as shown in the output.

The second part of the output shows the estimates for intercept and slope. The value for slope is 0.1364. The 95% confidence interval estimate of the population slop coefficient is from 0.1116 to 0.1614. It concludes that a 1 unit increase in GDP growth will result in an increase between 0.1116 and 0.1614 units of productivity growth (Groebner, 2008). While this interval does not include the value 0, it concludes that the slope is non-zero (Jessop, 2010). It can be sure that there is a positive slope but are uncertain of the extent. Also, the value of standard error indicates the variation of the slope value from sample to sample. If the standard error of the slope is large, the value of slope will be variable. The regression output in figure 4 shows the standard error is 0.0126, so that, the slope values will be less variable and the sample regression lines will cluster closely around the true population line (Jessop, 2010).

Moreover, the very small P-value associated with the t test for significance. Given the level of significance =0.05, the decision of whether to reject can be made as

Reject , if p-value <

Because the p-value = 1.57E-18<=0.05, it rejects and conclude that there is a significant relationship between quarterly GDP growth and quarterly productivity growth (Anderson et al, 2003).

## Significance tests

The significance test of the correlation coefficient and the regression coefficient are developed from the sample data which point estimates of the population correlation coefficient and the true regression coefficients for the population that are subject to sampling error. Therefore, it needs a test procedure to determine whether the coefficient is statistically significant. Moreover, the test for the simple linear regression slope coefficient is equal to the test for the correlation coefficient. That is when the correlation between variable x and variable y is found significant, and then the regression slope coefficient will also be significant (Groebner et al, 2008).

## Correlation significance test

The correlation coefficient computed from 100 sample data is a point estimate of the population correlation coefficient. Although the correlation coefficient of 0.7394 seems high, compare to 0, it is subject to sampling error. The formal hypothesis-testing procedure is needed to determine whether the linear relationship between GDP growth and productivity growth is significant.

The null and alternative hypotheses to be tested are:

(no correlation)

(correlation exists)

=0.05

represents the population correlation coefficient. It is must to test whether the sample data support or refute the null hypothesis. The test procedure utilizes the t-test statistic.

Test statistic for correlation

Where: t = number of standard errors r is from 0

r = sample correlation coefficient

n = sample size

df = n-2

The calculated t-value is

= 10.8719

Decision rule:

If t > t 0.025 = 1.9845, reject Ho

If t < -t 0.025 = -1.9845, reject Ho

Otherwise, do not reject Ho

Because 10.8719 > 1.9845, reject Ho. Based on the sample evidence, it concludes that there is a significant, positive linear relationship in the population between quarterly GDP growth and quarterly productivity growth.

## Significant test of the regression slope

Commonly, the sample regression slope is not likely equal to the true population slope. To determine if the population regression slope coefficient is 0 or not, the test of significance of the simple linear regression slope coefficient is used. A slope of 0 indicates that the relationship between x and y variables is not linear. That is, in this linear form, independent variable x will neither explain any variation in the dependent variable y, nor useful in predicting the dependent variable. If the linear relationship is significant, then the hypotheses of regression slope is 0 should be rejected. However, the estimated regression slope coefficient is calculated from sample data, it is subject to sampling error. Therefore, even though estimated regression slope coefficient is not 0, it is necessary to determine if the difference from 0 is greater than would mainly be attributed to sampling error (Groebner et al, 2008)

The null and alternative hypotheses to be tested at the 0.05 level of significance are:

=0.05

Simple linear regression test statistic for test of the significance of the slope

Where: = sample regression slope coefficient

= hypothesized slope (usually=0)

=estimator of the standard error of the slope

The calculated t is

==10.8719

Decision rule:

If , reject

If , reject

Otherwise, do not reject

Because of 10.8719 > 1.9845, null hypothesis should rejected and conclude that the true slope is not 0. Therefore, the simple linear relationship that utilizes the independent variable, quarterly GDP growth, is helpful to explain the variation in the dependent variable, quarterly productivity growth (Greobner, 2008).

## Conclusion

To find out the relation between quarterly GDP growth and quarterly productivity growth, the analysis of correlation, regression models and the test of significant are used. From the analysis results, it shows that quarterly GDP growth has the relation with quarterly productivity growth.

The correlation coefficient of these two variables is 0.7394; it indicates the strength of relation between them. Also, it is possible to forecast quarterly productivity growth by given the quarterly growth rate of GDP. However, the correlation coefficient only implies the tendency of the data. It is not support the cause and effect relationship. The Fisher's transformation is based on sample data to make an inference of the correlation coefficient value of a population. The 95% of confidence intervals for correlation coefficient of the population is from 0.6353 to 0.8171.

The linear regression model develops the equation of y = -0.1546 + 0.1365 x. The number of slope, which is 0.1365, represents when GDP growth increases per 1 unit, the mean value of productivity growth will be increased by 0.1365 units. From the summary output of regression analysis, the 100 sample size shows the risk of incorrectly concluding non-zero population correlation is lower and the relation between these two variables are significant. R square suggests the variation in the dependent variable can be explained by independent variable. The portion of productivity growth can be explained by GDP growth is 54.67%. Moreover, the number of slope is 0.1365; the 95% confidence interval forecast of the population slop coefficient is between 0.1116 and 0.1614. It shows a positive slope but unsure of the extent. The small number of standard error, which is 0.0126, suggests that the slope values will be less variable and the sample regression lines will come together around the true population line. The small number of p-value related with t test for significance, it concludes that there is a significant relationship between quarterly GDP growth and quarterly productivity growth.

From correlation significance test and the regression slope significant test, it comes out the same results which are a significant positive linear relationship between quarterly GDP growth with quarterly productivity growth. And the quarterly GDP growth is helpful to explain the variation in quarterly productivity growth.

## Reflection

This paper analyzes the relationship between quarterly GDP growth and quarterly productivity growth. The numbers of quarterly growth rate came from the comparison with the same quarter of previous year. Moreover, the sample data focused on manufacturing section in UK.

Research shows that the production industries, such as manufacturing, construction, distribution, hotels and catering sector and transport and communication, have more elastic in the downturn in terms of productivity growth. This partially reflects a more moderate drop in output growth in production sector; also, it takes more time to modify the arrangement of labour input than some other sectors (Myers, 2009). If the sample data came from other non-production industries, the result might different.

The coefficient of determination, which is 54.67%, shows the proportion of variation in productivity growth can be explained by GDP growth. Namely, there are other factors might influence the productivity growth. Productivity is the ratio of output to the input of labour. The hours worked is used to measure of labour input rather than the numbers of employees. The output is influenced by many elements such as the resources of capital equipment, new technologies, and management practices that are outside of worker's influence.

Moreover, the economic status might have different level of impacts on productivity growth. During economic recession, productivity growth would not go down as much as output growths in theory, because companies laid off the least productive employees first and retained the more productive employees. Likewise, in economic upturn, productivity growth is expected to be lower than output growth. Because when more employees are recruited, the skill and experience level of the incremental employees decline, and so does their potential of productive (Myers, 2009). The future research could emphasize on discovering other factors which are impacts on productivity growth and analyze the relationship among variables.

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