The role of technological progress in the economic process is a central theme in neoclassical growth models. Many economic studies that examine the factors influencing economic growth are based on the classic Solow growth model and its various adaptations.

The Solow model posits that a country's level of production is determined by capital, labor, and technology, which are exogenously determined as follows

$${Y}_{it}=F\left({A}_{it}, {K}_{it} , {L}_{it}\right),$$

where,\({ Y}_{it}\), \({A}_{it}, {K}_{it} and {L}_{it}\) are output, the level of technology, capital stock, and labor, respectively, in country 𝑖 at time t. The specific functional form adopted is the three-input, constant returns to scale, Cobb-Douglas production function with labor-augmenting technology as in Mankiw et al. (1992).

$${Y}_{it}= {{K}_{it}}^{\alpha }{{H}_{it}}^{\beta }({A}_{it}{L}_{it}{)}^{1-\alpha -\beta }$$

where, \({H}_{it}\) is now human capital in country \(i\) at time t, and \(\alpha\) and \(\beta\) are both positive fractions.

The evolution of physical and human capital is given by the following first-order differential equations.

$$\dot{\widehat{k}}={s}_{k}y-\left(n+g+\delta \right)\widehat{k}$$

$$\dot{\widehat{h}}={s}_{h}y-\left(n+g+\delta \right)\widehat{h}$$

where \({s}_{k}\) and \({s}_{h}\) are the marginal propensities to save for physical and human capital, g is the rate of technological progress, \(n\) denotes population growth, and \(\delta\) is the rate of depreciation, which we assume to be the same for physical and human capital for simplification. \(\widehat{k}\) and \(\widehat{h}\) are physical and human capital per *effective* labor, and \(\dot{\widehat{k}}\) and \(\dot{\widehat{h}}\) are their respective first derivatives with respect to time.

Their steady state value is given as:

$${k}^{*}= {\left[\frac{{s}_{k}^{1-\beta } {s}_{h}^{\beta }}{n+g+\delta }\right]}^{\frac{1}{1-\alpha -\beta }}$$

$${h}^{*}={\left[\frac{{s}_{h}^{1-\alpha } {s}_{k}^{\alpha }}{n+g+\delta }\right]}^{\frac{1}{1-\alpha -\beta }}$$

\({k}^{*}\) and \({h}^{*}\) are the steady state values of physical and human capital respectively.

Substituting the steady state values of physical and human capital into the production function, assuming that the marginal propensity to save for physical capital are the same, and taking the natural logarithm of both sides, we obtain the following equation.

$$\text{ln}y=\text{ln}{A}_{0}+gt+ \frac{\alpha +\beta }{1-\alpha -\beta }\text{ln}\left(n+g+\delta \right)+\frac{\alpha +\beta }{1-\alpha -\beta }\text{ln}s$$

where \(y\) is GDP per capital.

We extend this structural equation to include other factors affecting GDP per capital, as identified by the growth literature.

$$\text{ln}y=\text{ln}{A}_{0}+gt+ \frac{\alpha +\beta }{1-\alpha -\beta } \text{l}\text{n}(n+g+\delta )+ \frac{\alpha +\beta }{1-\alpha -\beta }\text{ln}s+{\Phi }X$$

\(X\) is a vector of control variables that affect GDP per capita, and \({\Phi }\) is the vector of coefficients. One of these controls is an index of digitization.

The previous equation can be estimated using regression analysis as follows.

$$\text{ln}{GDP per Capita}_{it} =\beta o+{\beta }_{4}Dig{itzation}_{it}+{\beta }_{16}{Domestic Credit}_{it}+{\beta }_{14}{Savings}_{it}+{\beta }_{1}{Capital Formation}_{it}+{\beta }_{2}{Human Capital}_{it}+{\beta }_{7}\text{ln}{Trade}_{it}+{\beta }_{15}{Institutions}_{it}+{\mu }_{i}+ {ϵ}_{it}$$

The regression equation includes a time trend to eliminate the effect of trends in the dependent and independent variables and mitigate the occurrence of spurious regression.

## Data, Methodology, and Identification

## Data

This paper relied on data extracted from the World Bank’s World Development Indicators and Penn World Table Version 10. The study used an unbalanced panel data on 33 Sub-Saharan African countries for the period from 1996 to 2019. The selection of the study period and countries depended on the availability of data. Table 1 shows the list of variables used, data source and their descriptions and measurements.

Table 1

Variable | Definition | Measurement |
---|

Real Gross Domestic Product (GDP) per capita | GDP per capita is gross domestic product divided by midyear population. Data are in constant 2015 U.S. dollars. | World Bank, World Development Indicators |

Gross fixed capital formation | Gross fixed capital formation (formerly gross domestic fixed investment) includes transactions like land improvements; plant, machinery, and equipment purchases; and the construction of roads, railways. Data are as expressed % of GDP. | World Bank, World Development Indicators |

Gross domestic saving | Gross savings are calculated as gross national income less total consumption, plus net transfers. Data are as expressed % of GDP. | World Bank, World Development Indicators |

Trade | Trade is the sum of exports and imports of goods and services measured as a share of gross domestic product. | World Bank, World Development Indicators |

Human capital index | Human capital index, based on years of schooling and returns to education; | Penn World Table version 10 |

Financial development | Financial resources provided to the private sector by financial corporations, such as through loans, purchases of nonequity securities, and trade credits and other accounts receivable, that establish a claim for repayment. Data are as expressed % of GDP. | World Bank, World Development Indicators |

Institutional quality index | An index created as the first principal component of the following variables: political stability, voice and accountability, government effectiveness, regulatory quality, rule of law and control of corruption. | Worldwide Governance Indicators |

Digitization index | An index created as the simple mean of individuals using the internet (% of population) and mobile cellular subscriptions (per 100 people) | World Bank, World Development Indicators |

Table 2 presents the descriptive statistics of the variables used in the formal regression analyses. The mean GDP per capita is $2,034 in constant 2015 USD. Our measure of digitization index is created as the simple mean of the percentage of population using internet and the number of mobile users per 100 people. Individually, the average of internet users and mobile cellular users are 8.5 and 38.9 respectively. The average of the digitization index is 23.8. The average domestic credit, gross domestic saving, gross capital formation, and international trade are 18.2%, 17.6%, 21.2% and 68.5%, respectively.

Table 2

Summary Statistics and Pairwise Correlations

| GDP per capita | Digitization Index | Domestic Credit | Gross Domestic Saving | Gross Capital Formation | Human Capital Index | Trade | Institution Quality Index |
---|

Mean | 2034.522 | 23.769 | 18.245 | 17.634 | 21.229 | 1.735 | 68.476 | 0 |

Std. Dev. | 2686.668 | 26.435 | 21.821 | 11.532 | 9.132 | 0.425 | 35.214 | 2.214 |

Min | 234.709 | 0 | 0 | -19.903 | -2.424 | 1.053 | 1.219 | -3.484 |

Max | 16992.033 | 130.158 | 142.422 | 57.85 | 81.021 | 2.939 | 225.023 | 5.587 |

Pairwise Correlations |

GDP Per Capita | 1 | | | | | | | |

Digitization Index | 0.447* | 1 | | | | | | |

Domestic Credit | 0.395* | 0.451* | 1 | | | | | |

Gross Domestic Saving | 0.238* | 0.086* | 0.055 | 1 | | | | |

Gross Capital Formation | 0.264* | 0.234* | 0.028 | 0.617* | 1 | | | |

Human Capital Index | 0.661* | 0.533* | 0.465* | 0.270* | 0.167* | 1 | | |

Trade | 0.611* | 0.315* | 0.171* | 0.230* | 0.393* | 0.406* | 1 | |

Institution Quality Index | 0.439* | 0.361* | 0.572* | 0.193* | 0.129* | 0.444* | 0.368* | 1 |

*** p < 0.01, ** p < 0.05, * p < 0.1; Countries covered by the study: Angola, Burundi, Benin, Burkina Faso, Botswana, Cote d’Ivoire, Cameroon, Dem. Rep. of Congo, Republic of Congo, Gabon, Ghana, The Gambia, Kenya, Lesotho, Madagascar, Mali, Mozambique, Mauritania, Mauritius, Namibia, Niger, Nigeria, Rwanda, Sudan, Senegal, Sierra Leone, Eswatini, Togo, Tanzania, Uganda, South Africa, Zambia, Zimbabwe |

## Methodology and Identification

We employed several estimation techniques to estimate the relationship between GDP per capita and digitization, including pooled Ordinary Least Squares (OLS), fixed effects, and two-stage least squares instrumental variable (2SLS IV) methods. However, it is important to acknowledge the limitations and potential biases associated with each of these techniques. The estimates obtained from the pooled OLS regression may be biased due to the neglect of the panel structure of the data. This method fails to consider country-specific factors that are unique to each country but are included in the error term. As a result, it violates the zero-conditional mean assumption of OLS.

To address this issue, we employ the fixed effects regression, which eliminates time-invariant country-specific attributes and represents an improvement over the pooled OLS approach. However, the fixed effects regression does not fully account for potential endogeneity problems arising from omitted time-varying variables and reverse causality between GDP per capita and digitization. It is worth noting that countries with higher GDP per capita tend to be more digitized, and vice versa. Therefore, the fixed effects regression results may also be subject to endogeneity concerns.

To overcome these endogeneity issues, we employ instrumental variables techniques, specifically the two-stage least squares instrumental variable (2SLS IV) estimation. By identifying and utilizing appropriate instrumental variables, we can address the potential endogeneity biases and obtain more reliable estimates. In this IV approach, we instrument the level of digitization (\({digitzation}_{it}\)) with the first difference of digitization (\({digitzation}_{it}-{digitzation}_{it-1})\). This method of generating internal instruments follows the approach proposed by Anderson and Hsiao (1982). By construction, \({digitzation}_{it}-{digitzation}_{it-1}\) is correlated with \({digitzation}_{it}\), making it a relevant instrument. The exclusion restriction, which is essential for instrument validity, implies that changes in digitization impact GDP per capita solely through their effect on the level of digitization. While changes in digitization could potentially affect the *growth* rate of GDP per capita, there are no established theories linking changes in digitization to the *level* of GDP per capita. Therefore, conditional on satisfying the diagnostic tests we conduct, the instrument allows us to address the endogeneity issue present in the pooled OLS and fixed effects regressions. Thus, by utilizing this instrumental variable approach, we can overcome the endogeneity problem and establish a causal relationship between digitization and GDP per capita.

## Diagnostic Tests, Results, and Discussion

## Diagnostic Tests

Before presenting the main regression results, we perform three diagnostic tests: tests for under-identification, tests for weak identification, and tests for weak-instruments-robust inference.

Table 3

Under Identification | Test-Statistic | P-Value | |
---|

Kleibergen-Paap rk LM Test | Chi-sq(1) = 16.78 | 0.0000 | |
---|

Kleibergen-Paap rk Wald Test | Chi-sq(1) = 20.21 | 0.0000 | |
---|

**Weak Identification** | | | |

**Kleibergen-Paap rk Wald F-Statistic** | **19.85** | | |

**Stock-Yogo Weak ID test critical Values** | | | |

**10% maximal IV size** | **16.38** | | |

**15% maximal IV size** | **8.96** | | |

**20% maximal IV size** | **6.66** | | |

**25% maximal IV size** | **5.53** | | |

**Weak Instrument Robust Inference** | | | |

**Anderson-Rubin Wald Test** | **F(1,507) = 12.61** | **0.0004** | |

**Anderson-Rubin Wald Test** | **Chi-sq(1) = 12.83** | **0.0003** | |

**Stock-Wright LM S statistic** | **Chi-sq(1) = 12.17** | **0.0005** | |

Note: Under Identification test: H0: Matrix of reduced form coefficients is under identified; Ha: matrix has full rank (identified) |

Weak Identification Test: H0: Equation is weakly identified; Ha: H0 is not true.

Weak-Instrument Robust Inference: H0: The orthogonality conditions are valid.

The under-identification test, specifically the heteroskedasticity-robust Kleibergen-Paap rk test, examines the correlation between the instrument (the change in digitization) and the endogenous regressor (the level of digitization). The null hypothesis for this test suggests that the equation is under-identified, indicating that the matrix of reduced form coefficients has reduced rank. Rejection of the null hypothesis indicates that the matrix has full rank, and the system is identified. The results of this test can be found in Table 3. Upon reviewing the table, we observe that the test statistics for both the LM and Wald versions are 16.78 and 20.21, respectively, with corresponding p-values of 0.0000 for both versions. Consequently, we strongly reject the null hypothesis, providing evidence that the equation is identified.

The identification test we employ, namely the Kleibergen-Paap rk F test, serves to examine not only the presence of instrument correlation with the endogenous regressor but also the strength of that correlation. It is important to note that weak instruments can significantly compromise the performance of estimators. In our analysis, the test statistic value obtained, which is 19.853 as presented in Table 3, surpasses the conventional critical values derived by Stock and Yogo (2005) for various maximal IV sizes, specifically 10%, 15%, 20%, and 25%. As a result, we reject the null hypothesis of weak instruments, affirming that the instruments used in our study exhibit strength and validity.

We conduct the final set of diagnostic tests, namely the Anderson-Rubin Wald test and the Stock-Wright S test, which are specifically designed for weak-instrument-robust inference. These tests examine the joint insignificance of the coefficients associated with the endogenous regressors and the validity of the overidentifying orthogonality restrictions. In our model, we have one endogenous regressor and one instrument, resulting in exact identification without overidentification. Additionally, since our focus lies on one specific endogenous regressor, digitization, these tests essentially evaluate the significance of that particular regressor. Upon analyzing the results presented in Table 3, we observe that both tests reject the null hypothesis with corresponding p-values 0.0004 and 0.0005, providing evidence for the significance of our endogenous regressor. This outcome supports our research hypothesis and underscores the importance of the endogenous variable in explaining the relationship under investigation.