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# Empirical analysis of the economic absolute income hypothesis based on mathematical statistics

###### Przyjęty: 17 Apr 2022
Informacje o czasopiśmie
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Hypothesis testing in absolute choice

Selection model is an important part of the modeling analysis, economics in looking for the true model, in the design of multiple models assume that are separate from the actual setting, then according to this problem is the separation of absolute choice model, it has the identity and the separation of assumption in mathematical statistics, thus can take advantage of each other during solve processing model is analyzed. The available methods are mainly divided into two types: on the one hand, it refers to the improved LR test; on the other hand, it refers to the embedding coefficient λ to transform the problem into a conventional parameter for testing. This paper mainly discusses the connection and characteristics of the two in the application period, and shows the special function of alternative hypothesis H1 in these test methods, as well as its specific relationship with prior information. At this point, we should clearly recognize the relationship between mathematical tools, economic theoretical knowledge, statistics and other contents as shown in the figure 1 below[1]:

Definition 1

In the study, Pesaran et al. conducted Cox tests on the linear econometric model and could propose two separate PDF forms. The actual single econometric model formula is shown as follows: $H0:fi(y,θi|x) θi∈ΘiH1:fj(y,θj|x) θj∈Θj$ \eqalign{ & {H_0}:{f_i}\left( {y,{\theta _i}|x} \right)\,\,\,\,\,\,\,\,\,\,\,{\theta _i} \in {\Theta _i} \cr & {H_1}:{f_j}\left( {y,{\theta _j}|x} \right)\,\,\,\,\,\,\,\,\,\,\,{\theta _j} \in {\Theta _j} \cr}

Theorem 1

In the above formula, Y represents the endogenous variable and X represents the exogenous variable, which does not meet the following conditions: $θi*∈Θ, as well as θj*∈Θ$ \theta _i^* \in \Theta ,\,\,\,\,\,{as}\,\,\,\,\,\,{well}\,\,{as}\,\,\theta _j^* \in \Theta

Thus $f0(*,θi*|x)=f1(*,θj*|x)$ {f_0}\left( {*,\theta _i^*|x} \right) = {f_1}\left( {*,\theta _j^*|x} \right) was established.

Assume that $θ^i$ {\hat \theta _i} , $θ^j$ {\hat \theta _j} both represent MLE under H0 and H1, and the actual definition formula is as follows: $Lk=Lk(θk)=∑i=1Nlnfk(yt,θk|x)(k=i,j,N is the sample size),Lij(θ^i,θ^j)=Li(θ^i)−Lj(θ^j)$ {L_k} = {L_k}\left( {{\theta _k}} \right) = \sum\limits_{i = 1}^N {\ln {f_k}\left( {{y_t},{\theta _k}|x} \right)\left( {k = i,j,N\,\,is\,\,the\,\,sample\,size} \right),{L_{ij}}\left( {{{\hat \theta }_i},{{\hat \theta }_j}} \right) = {L_i}\left( {{{\hat \theta }_i}} \right) - {L_j}\left( {{{\hat \theta }_j}} \right)}

The n the results can be verified. Under the condition of H0, the following formula can be obtained: $Ti=Lij(θ^i,θ^j)−E^i(Lij(θ^i,θ^ij))≈N(0,V(Ti))$ {T_i} = {L_{ij}}\left( {{{\hat \theta }_i},{{\hat \theta }_j}} \right) - {\hat E_i}\left( {{L_{ij}}\left( {{{\hat \theta }_i},{{\hat \theta }_{ij}}} \right)} \right) \approx N\left( {0,V\left( {{T_i}} \right)} \right)

In the formula, Êr(•) refers to the expected operation distributed in $fi(y,θ^i|x)$ {f_i}\left( {y,{{\hat \theta }_i}|x} \right) , V (Tt) refers to the variance of Tt on H0, and represents a consistent estimate of θij, which meets the following conditions: $plimθ^j=θij$ p\lim {\hat \theta _j} = {\theta _{ij}}

Proposition 2

Considering T1 as a statistic, a COX test can be constructed to select the model. As another way to solve the problem, embedded verification has two choices in practical application. When the econometric model is in PDF form, the model formula can be obtained as follows: $f(y,θi,θj,λ|x)=fiλ(y,θj|x)fj1−λ(y,θj|x)[∫fiλ(y,θj|x)fj1−λ(y,θj|x)dy]−1$ f\left( {y,{\theta _i},{\theta _j},\lambda |x} \right) = f_i^\lambda \left( {y,{\theta _j}|x} \right)f_j^{1 - \lambda }\left( {y,{\theta _j}|x} \right){\left[ {\int {f_i^\lambda \left( {y,{\theta _j}|x} \right)f_j^{1 - \lambda }\left( {y,{\theta _j}|x} \right)dy} } \right]^{ - 1}}

The actual assumptions are: $H0:λ=1;H0:λ≠1$ {H_0}:\lambda = 1;{H_0}:\lambda \ne 1

At this time, the result verification found that under the condition of H0, the formula of the actual statistic was: $T˜i=∑i=1N[ξit−ξjt−Ei(ξi−ξj)−covi(ξi−ξj∨ξi).∨ξit/vari(∨ξi)]$ {\tilde T_i} = \sum\limits_{i = 1}^N {\left[ {{\xi _{it}} - {\xi _{jt}} - {E_i}\left( {{\xi _i} - {\xi _j}} \right) - {{{\mathop{\rm cov}} }_i}\left( {{\xi _i} - {\xi _j} \vee {\xi _i}} \right). \vee {\xi _{it}}/{{{\mathop{\rm var}} }_i}\left( { \vee {\xi _i}} \right)} \right]}

There is a gradual normal distribution trend of zero mean during the overall operation, and the specific results are shown as follows: $ξkt=lnfkk(yt,θk|xt),ξk=lnfk(y,θk|x),k=i,j∇ξit=∂ξit/∂θi,∇ξi=∂ξi/∂θi$ \eqalign{ & {\xi _{kt}} = \ln f{k_k}\left( {{y_t},{\theta _k}|{x_t}} \right),{\xi _k} = \ln {f_k}\left( {{y},{\theta _k}|{x}} \right),k = i,j \cr & \nabla {\xi _{it}} = \partial {\xi _{it}}/\partial {\theta _i},\nabla {\xi _i} = \partial {\xi _i}/\partial {\theta _i} \cr}

In the above formula, E1 represents the expectation of fi(y, θi|x), cov1 represents the covariance of fi(y, θi|x), and VAR1 represents the variance operation of fi(y, θi|x).

When the econometric model uses the equation model for calculation and analysis, the following formula can be obtained: $H0:Y=Pi(x,θi)+εi.εi∼N(0,σi2)H1:Y=Pj(x,θj)+εj,εj∼N(0,σj2)$ \eqalign{ & {H_0}:Y = {P_i}\left( {x,{\theta _i}} \right) + {\varepsilon _i}.{\varepsilon _i}\sim N\left( {0,\sigma _i^2} \right) \cr & {H_1}:Y = {P_j}\left( {x,{\theta _j}} \right) + {\varepsilon _j},{\varepsilon _j}\sim N\left( {0,\sigma _j^2} \right) \cr}

Lemma 3

Based on this, the embedded analysis can obtain the following model: $y=(1−λ)Pi(x,θi)+λPi(x,θi)+ε$ y = \left( {1 - \lambda } \right){P_i}\left( {x,{\theta _i}} \right) + \lambda {P_i}\left( {x,{\theta _i}} \right) + \varepsilon

The actual assumptions are: $H0:λ=0;H1:λ≠0$ {H_0}:\lambda = 0;{H_1}:\lambda \ne 0

As the above model formula is nonlinear under the condition of θ = (θvθy, λ), it is difficult to be directly tested in the condition of H0, so it can be regarded as approximate value. Plug $MLEθ^j$ MLE{\hat \theta _j} into the equation model, and the linearized formula can be obtained, as shown below: $y−P^i=λ(P^j−P^i)+xTS^i+ε$ y - {\hat P_i} = \lambda \left( {{{\hat P}_j} - {{\hat P}_i}} \right) + {x^T}{\hat S_i} + \varepsilon

Conjecture 5

In the above formula, the condition of $P^k=Pk(x,θ^k),k=i,j$ {\hat P_k} = {P_k}\left( {x,{{\hat \theta }_k}} \right), k = i,j ; $S^t=∂PI(x,θ^t)/∂θt$ {\hat S_t} = \partial {P_I}\left( {x,{{\hat \theta }_t}} \right)/\partial {\theta _t} is met, and represents the regression coefficient of Ŝi before AR. At the same time, the above model formula is linear, and LS estimation can also be performed. In addition, significance test analysis of αT, parameter λ can be proposed by using the above hypothesis conditions to judge whether the proposed hypothesis conditions are valid.

In order to construct the test using $T˜i$ {\tilde T_i} , it is necessary to estimate and analyze θ I and θj under the condition of H0. The former is $θ^i$ {\hat \theta _i} , while the latter is difficult to estimate directly from the above formula. For H0, to satisfy the requirement of $plimθ^j=θij$ p\lim {\hat \theta _j} = {\theta _{ij}} , we need to replace θj with $θ^ij$ {\hat \theta _{ij}} . Plug the relevant values into the statistical formula and meet the requirement of $∑i=1N∨ζ1i(θ^i)=0$ \sum\limits_{i = 1}^N { \vee {\zeta _{1i}}\left( {{{\hat \theta }_i}} \right) = 0} , and you will eventually get the following formula: $T˜i=⌊Li(θ^i)−Lj(θ^ij)−E^i⌊Li(θ^i)−Lj(θ^ij)⌋⌋$ {\tilde T_i} = \left\lfloor {{L_i}\left( {{{\hat \theta }_i}} \right) - {L_j}\left( {{{\hat \theta }_{ij}}} \right) - {{\hat E}_i}\left\lfloor {{L_i}\left( {{{\hat \theta }_i}} \right) - {L_j}\left( {{{\hat \theta }_{ij}}} \right)} \right\rfloor } \right\rfloor

Therefore, in the above formula, except for the substitution of $θ^ij$ {\hat \theta _{ij}} in the first parenthesis by $θ^i$ {\hat \theta _i} , the right side of the equation is identical, but under the condition of H0, it meets the requirement of $θ^i−θ^ij=op(1)$ {\hat \theta _i} - {\hat \theta _{ij}} = {o_p}(1) . Therefore, the embedding test and COX test obtained by the statistical model formula are asymptotically equivalent, and the corresponding logarithmic likelihood function model is shown as follows: $L(θiθjλ)=λLi(θi)+(1−λ)Lj(θj)−ln⌊∫fiλ(y,θt|x)fj1−λ(y,θj|x)dy⌋$ L\left( {{\theta _i}{\theta _j}\lambda } \right) = \lambda {L_i}\left( {{\theta _i}} \right) + (1 - \lambda ){L_j}\left( {{\theta _j}} \right) - \ln \left\lfloor {\int {f_i^\lambda \left( {y,{\theta _t}|x} \right)f_j^{1 - \lambda }\left( {y,{\theta _j}|x} \right)dy} } \right\rfloor

Example 6

Under weak conditions, $θ^ij$ {\hat \theta _{ij}} is used for formula analysis, and the test statistics of the above model are: $T¯i2/V(T¯i)$ \bar T_i^2/V\left( {{{\bar T}_i}} \right)

And meet the following conditions: $T¯i=Lij(θ^i,θ^j)−E^i⌊Lij(θ^i,θ^j)⌋$ {\bar T_i} = {L_{ij}}\left( {{{\hat \theta }_i},{{\hat \theta }_j}} \right) - {\hat E_i}\left\lfloor {{L_{ij}}\left( {{{\hat \theta }_i},{{\hat \theta }_j}} \right)} \right\rfloor

It follows that the actual asymptotic is equal to the square of the standardized Score test statistic. By using $θ^i$ {\hat \theta _i} for formula analysis, the actual Score statistic will be transformed into the square of the standardized embedded test statistic.[2,3]:

According to the model obtained from the above research, the relationship of embedded test is analyzed. The actual equation model is as follows: $fi(y,θiσi2|x)=(2πσi2)−1.2exp(−(2σi2)−1(y−fi(x,θi))2)fj(y,θjσj2|x)=(2πσj2)−1.2exp(−(2σj2)−1(y−fj(x,θj))2)$ \eqalign{ & {f_i}\left( {y,{\theta _i}\sigma _i^2|x} \right) = {\left( {2\pi \sigma _i^2} \right)^{ - 1.2}}\exp \left( { - {{\left( {2\sigma _i^2} \right)}^{ - 1}}{{\left( {y - {f_i}\left( {x,{\theta _i}} \right)} \right)}^2}} \right) \cr & {f_j}\left( {y,{\theta _j}\sigma _j^2|x} \right) = {\left( {2\pi \sigma _j^2} \right)^{ - 1.2}}\exp \left( { - {{\left( {2\sigma _j^2} \right)}^{ - 1}}{{\left( {y - {f_j}\left( {x,{\theta _j}} \right)} \right)}^2}} \right) \cr}

And can be incorporated into the following models: $f(y,θiσi2σj2|x)=(2πσi2)−1.2exp(−(2σ2)−1(y−u)2)$ f\left( {y,{\theta _i}\sigma _i^2\sigma _j^2|x} \right) = {\left( {2\pi \sigma _i^2} \right)^{ - 1.2}}\exp \left( { - {{\left( {2{\sigma ^2}} \right)}^{ - 1}}{{\left( {y - u} \right)}^2}} \right)

Note 7

Among them, $σ2=σi2σj2/[(1−λ)σj2+λσi2]u=[(1−λ)σ2/σj2]fj(x,θj)+[λσ2/σi2]fi(x,θi)$ \eqalign{ & {\sigma ^2} = \sigma _i^2\sigma _j^2/\left[ {\left( {1 - \lambda } \right)\sigma _j^2 + \lambda \sigma _i^2} \right] \cr & u = \left[ {\left( {1 - \lambda } \right){\sigma ^2}/\sigma _j^2} \right]{f_j}\left( {x,{\theta _j}} \right) + \left[ {\lambda {\sigma ^2}/\sigma _i^2} \right]{f_i}\left( {x,{\theta _i}} \right) \cr}

The actual regression equation model is as follows: $y=[λσj2/((1−λ)σj2+λσi2)]fi(x,θi)+[(1−λ)σi2/(1−λ)σj2+λσi2]fj(x,θj)+εε∼N(0,σ2)$ \eqalign{ & y = \left[ {\lambda \sigma _j^2/\left( {\left( {1 - \lambda } \right)\sigma _j^2 + \lambda \sigma _i^2} \right)} \right]{f_i}\left( {x,{\theta _i}} \right) + \left[ {\left( {1 - \lambda } \right)\sigma _i^2/\left( {1 - \lambda } \right)\sigma _j^2 + \lambda \sigma _i^2} \right]{f_j}\left( {x,{\theta _j}} \right) + \varepsilon \cr & \varepsilon \sim N\left( {0,{\sigma ^2}} \right) \cr}

It can be seen from this that the above formula can be transformed into the econometric model equation proposed in previous studies under the condition that σi = σj = σ0 is met. In other words, in the Gaussian model, if the condition σiσj is met, then the two embedding tests studied above are identical; If the condition σiσj is met, it means that the tests of λ and $λα=λσj2/[(1−λ)σj2+λσi2]$ {\lambda _\alpha } = \lambda \sigma _j^2/\left[ {\left( {1 - \lambda } \right)\sigma _j^2 + \lambda \sigma _i^2} \right] are equivalent.

Common mathematical statistics problems require that both the original hypothesis (H0) and the alternative hypothesis (H1) in hypothesis testing have all the possibilities of the problem. Therefore, if the original hypothesis is rejected, the alternative hypothesis must be accepted, and vice versa. In other words, the alternative hypothesis has a certain degree of substitution, after the original hypothesis has been rejected, the alternative hypothesis will take over and become true.

Open Problem 8

It should be noted that when testing the selection model based on the hypothesis, the substitutability of the alternative hypothesis will be affected, because there are many models that can be selected, so the alternative hypothesis will have a new change. According to the structure analysis of the single econometric model constructed in this study, under the condition that it belongs to the Kuopmen-Darmois family function, the test statistics will be presented intuitively, and the specific form is as follows:[4,5] $Si=r(θ^j)−r(θ^ij)$ Si = r\left( {{{\hat \theta }_j}} \right) - r\left( {{{\hat \theta }_{ij}}} \right)

From the perspective of practice, there are two main functions of alternative hypothesis during model selection: on the one hand, if the prior information makes people not fully believe that it is true, the alternative hypothesis only plays an indicative role; If the prior information proves that it is likely to be true, the alternative hypothesis not only has an indicative function, but also considers the possibility of using FJ to replace FI after the original hypothesis is rejected.

Analysis of mathematical statistics in economics

Accurate understanding of assumptions plays a positive role in the empirical analysis of economic absolute income hypothesis. As a social discipline, economics needs to effectively integrate statistical data, mathematical statistics, qualitative and quantitative analysis and other contents in practical research. Among them, regression analysis method, as the main form of econometric analysis, will put forward assumptions about unknown functions and various forms of statistical distribution. If you do not want to carry out hypothetical analysis, you can directly use semi-parametric and non-parametric methods to conduct research. The effective integration of regression analysis method and factor analysis can be called path analysis. Econometric model includes discrete selection model, quantile regression, multiple linear regression, linear regression model and so on. After the EM algorithm was put forward in the 1970s, the mathematical statistics method was put forward by researchers in the integration innovation and development of computer technology. Although variance analysis and regression analysis in the traditional sense are still widely used, the practical application effect is far inferior to that of mathematical statistics.[6,7]

Analysis of absolute income hypothesis

The absolute income hypothesis proposed by Keynes effectively integrates consumption expenditure and income level, and puts forward that the absolute income level affects the actual consumption level in practice verification. In essence, absolute income hypothesis mainly involves the following contents: First, actual consumption expenditure is a stable function with real income as the core; Second, the income level represents the absolute actual income level at the present stage; Third, the marginal propensity to consume is positive, but lower than 1. Fourth, marginal consumption will change with the change of real income level, and the two are inversely proportional; Fifth, marginal consumption shows a propensity level lower than average consumption. To transform real problems into economic models is to study the mathematical and physical processes of the economic absolute income hypothesis by combining mathematical statistics, and the detailed explanation is as follows:

Example 7

This paper mainly analyzes the relationship between residents' income and consumption level in the early 21st century, and constructs the unitary regression model as follows: $CONSP=α0+α1GDPP+μ$ CONSP = {\alpha _0} + {\alpha _1}GDPP + \mu

In the above formula, GDPP refers to China's per capita GDP and CONSP refers to China's residents' consumption level, which is a constant term. According to the theoretical analysis of absolute income hypothesis, this value is greater than 0 and less than 1.

In order to further verify the validity of this hypothesis, relevant data information should be collected first and data pretreatment should be done well. Secondly, the scatter diagram should be used to compare and analyze the change relationship of the two indicators, Combined with the above analysis, it is found that there is a linear relationship between the income level and consumption of urban residents in China. Therefore, measurement software can be selected to represent the above analysis model in practical analysis and calculation. At the same time, regression analysis should be carried out in combination with the software. The specific results are shown in the following table. The actual regression equation is: $CON^SP=201.1189+0.3862GDPP(T Value) 13.5 53.47R2=0.99 F=2589.54 DW=0.55$ \eqalign{ & CO\hat NSP = 201.1189 + 0.3862GDPP \cr & \left( {T\,{Value}} \right)\,\,13.5\,\,53.47 \cr & {R^2} = 0.99\,\,\,\,\,\,F = 2589.54\,\,\,DW = 0.55 \cr}

Data results obtained by the least square method

 Source SS DF MS Number Of Obs=23F(1,21)=2859.54Prob>F=0.0000R-Squared=0.9927Adj R-Squared=0.9924Root MSE=33,265 Model 3164162.45 1 3164162.45 Residual 23237.0765 21 1106.52745 Total 3187399.52 22 144881.796 Consp Coef Std Err T P>|t| [95%Conf Interval] Gdpp .3861803 .0072217 53.47 0.000 .3711619 .4011987 _Cons 201.1189 14.88402 13.51 0.000 170.1659 232.0719 Durbin-Watson D-Statistic(2,23)=.5506366

Combined with the above results, it is found that the actual marginal propensity to consume is 0.3862, and the value is 53.47. In the significance test analysis, it is found that in the early 21st century, for every yuan of income increase of urban residents in China, there will be 0.39 yuan for consumption, which is in line with the absolute income hypothesis. As the DW value reaches 0.55, there may be error series, which requires researchers to conduct in-depth exploration of relevant values and models.

Empirical analysis

Combined with the analysis of the economic absolute income hypothesis studied above, it can be seen that the income proposed by Keynes is the absolute real income at this stage, and the actual calculation formula is as follows: $c=α+βy$ c = \alpha + \beta y

In the above formula, C refers to consumption and β refers to marginal propensity to consume. The corresponding calculation formula is as follows: $MPC=ΔcΔyorβ=ΔcΔy$ MPC = {{\Delta c} \over {\Delta y}}or\beta = {{\Delta c} \over {\Delta y}}

In the above formula, Y represents income, and α refers to the indispensable part of spontaneous consumption. In other words, if income is zero, basic living consumption must be achieved, and the product between the two refers to the consumption generated by income. It can be seen that the calculation formula of absolute real income has certain economic meaning.

This paper makes an empirical analysis of absolute income hypothesis from the perspective of mathematical statistics, and specifically studies the income level and consumption expenditure of residents in China, the United States, The United Kingdom and India. The data obtained are from statistical databases of various countries and international statistical yearbooks. Based on the original data collected, to determine the level of national income per capita, a calculation based on the GDP deflator is required. This paper studies the year 2010 as the benchmark and makes a general statistical analysis on the basis of grasping relevant exchange rates, thus obtaining the per capita consumption expenditure level of the four countries. Combined with the above analysis, it can be found that the gap between China and India is not large, while the US and UK are also very close, so it can be seen that China and India are more backward than the UK and US.

According to the analysis of the per capita consumption level of the above research, after obtaining relevant data, the panel data analysis method is used to construct the research model as shown below, and the software Eviews5.1 is used for simulation analysis, and the results can be obtained as shown in the following table 2[8,9]: $cs=c+c1+c2+c3+β1y1+β2y2+β3y3+β4y4+ui$ cs = c + {c_1} + {c_2} + {c_3} + {\beta _1}{y_1} + {\beta _2}{y_2} + {\beta _3}{y_3} + {\beta _4}{y_4} + {u_i}

Model research results

Public intercept The coefficient of Intercept
The United States China India Britain The United States China India Britain
−788.5 0.746268 0.338316 0.465463 0.723108 −1149.727 904.4732 869.8939 −624.6401
−0.94514 9.903561 0.432146 0.205649 9.877384
0.3486 0 0.6673 0.8378 0

In the table analysis above, the penultimate row refers to the value of t, and the last row refers to the value of P. Through comparative analysis, it is found that in terms of independent consumption level and marginal consumption tendency, the gap between China and India and the UK and the US is very significant, while the overall level between the UK and the US is more similar, which further verifies the colorful expected results proposed in the analysis of this paper. The actual indicators are as follows:

Index analysis

R-Squared Adjusted R-Squared S.E.Of Regression Sum-Squared-Resid Log-Likelibood Durbin-Watson Stat F-Statistic
0.992482 0.991542 909.5101 46323679 −522.565 1.27245 0

Combined with the above table 4 analysis, it is found that the model has a strong fitting effect and can intuitively present the income differences between China and India and between the UK and the US. At this time, to accurately study the changes of residents' income and consumption in China, we can analyze the data of each country separately. Since Both China and India are developing countries, the actual intercept does not change, and the slope coefficient will change with individual changes. The corresponding function form is shown as follows: $CS=c+βiY1+Y2+ui$ CS = c + {\beta _i}{Y_1} + {Y_2} + {u_i}

Actual design is shown in the following table:

Intercept design of China and India

intercept The coefficient of
China India
107.3149 0.347539 0.417531
5.892645 16.63877 11.56917
0 0 0

In the table above, the penultimate row refers to the value of t, and the last row refers to the value of P. Based on this analysis, it is found that the overall development level and resident income of China and India are similar, no matter the propensity level of marginal consumption or the level of independent consumption, which further verifies the apriori expected results of absolute income hypothesis.

In addition, the development of China and India should be compared with other information shown in the table below. The final result shows that China's spontaneous consumption level is higher than India's, while India's marginal propensity to consume is higher than China's. According to the comparison results of China and India, the model shown below is also used to analyze Britain and America. The actual intercept design is shown in the following table 5: $CS=c+βiY1+Y2+ui$ CS = c + {\beta _i}{Y_1} + {Y_2} + {u_i}

Other information

R-Squared Adjusted R-Squared S.E.Of Regression Sum-Squared-Resid Log-Likelibood Durbin-Watson Stat P(F-Statistic)
0.916297 0.907328 27.69846 21481.73 −149.554 1.154365 0

Intercept design of UK and US

Intercept System
The United States Britain
−1597.671 0.735813 0.730557
−0.793540 0.735813 11.75955
0.4339 0 0

The final results show that there is a great difference in the level of discretionary consumption expenditure between the two countries, while the actual marginal propensity to consume is consistent. A comparative analysis of the two models shows that the intercepts of China and India are positive, while those of the UK and the US are negative. Because the intercept refers to the content of autonomous consumption expenditure, which is expressed as individual payment in developing countries, and refers to the welfare expenditure of individuals by the government in developing countries, it is clear that there are great differences in welfare treatment between the UK and the US. In terms of actual marginal propensity to consume, China and India are lower than the United States and The United Kingdom, which is the most striking difference between developed and developing countries.

Conclusion

To sum up, absolute income hypothesis, as the most unique theoretical knowledge in the employment theory system proposed by Keynes, plays a positive role in the research of economics. Therefore, the empirical analysis of relevant hypotheses based on mathematical statistics should not only master the application methods, but also select a model based on practical research experience, and accumulate more valuable information, so as to provide effective basis for empirical analysis.

#### Intercept design of UK and US

Intercept System
The United States Britain
−1597.671 0.735813 0.730557
−0.793540 0.735813 11.75955
0.4339 0 0

#### Intercept design of China and India

intercept The coefficient of
China India
107.3149 0.347539 0.417531
5.892645 16.63877 11.56917
0 0 0

#### Other information

R-Squared Adjusted R-Squared S.E.Of Regression Sum-Squared-Resid Log-Likelibood Durbin-Watson Stat P(F-Statistic)
0.916297 0.907328 27.69846 21481.73 −149.554 1.154365 0

#### Data results obtained by the least square method

 Source SS DF MS Number Of Obs=23F(1,21)=2859.54Prob>F=0.0000R-Squared=0.9927Adj R-Squared=0.9924Root MSE=33,265 Model 3164162.45 1 3164162.45 Residual 23237.0765 21 1106.52745 Total 3187399.52 22 144881.796 Consp Coef Std Err T P>|t| [95%Conf Interval] Gdpp .3861803 .0072217 53.47 0.000 .3711619 .4011987 _Cons 201.1189 14.88402 13.51 0.000 170.1659 232.0719 Durbin-Watson D-Statistic(2,23)=.5506366

#### Model research results

Public intercept The coefficient of Intercept
The United States China India Britain The United States China India Britain
−788.5 0.746268 0.338316 0.465463 0.723108 −1149.727 904.4732 869.8939 −624.6401
−0.94514 9.903561 0.432146 0.205649 9.877384
0.3486 0 0.6673 0.8378 0

#### Index analysis

R-Squared Adjusted R-Squared S.E.Of Regression Sum-Squared-Resid Log-Likelibood Durbin-Watson Stat F-Statistic
0.992482 0.991542 909.5101 46323679 −522.565 1.27245 0

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